Geometry and Trigonometry Practice Questions — SAT — Page 5

Question 81Question →

A right triangle has acute angles $P$ and $R$ . The sine of angle $P$ is defined by $\frac{a}{13}$ and the cosine of angle $R$ is $\frac{5}{13}$ , where $a$ is a positive constant. What is the value of $\sin(R)$ ?

\frac{5}{13}

\frac{12}{13}

\frac{12}{5}

\frac{\sqrt{194}}{13}

Show answer & explanation

Answer: $\frac{12}{13}$

Answer

\frac{12}{13}

The correct answer is the value

\frac{12}{13}

. Since

P

and

R

are the acute angles of a right triangle, they are complementary, which means

\sin(P) = \cos(R) = \frac{5}{13}

. By using the Pythagorean identity

\sin^2(R) + \cos^2(R) = 1

, we can substitute the value of

\cos(R)

to get

\sin^2(R) + \left(\frac{5}{13}\right)^2 = 1

. Solving for

\sin(R)

yields

\sin^2(R) = 1 - \frac{25}{169} = \frac{144}{169}

. Since

R

is an acute angle, taking the positive square root yields

\sin(R) = \frac{12}{13}

.

Step-by-Step Solution

1

Apply the complementary angle identity to determine the value of the constant

a

.

Since

P

and

R

are the acute angles of a right triangle, they are complementary, meaning

P + R = 90^\circ

. The co-function identity states that

\sin(P) = \cos(R)

. Given

\cos(R) = \frac{5}{13}

, we have

\sin(P) = \frac{a}{13} = \frac{5}{13}

, which means

a = 5

.

This establishes the relationship between the trigonometric ratios of the two acute angles.

2

Use the Pythagorean identity to find

\sin(R)

.

Using the Pythagorean identity

\sin^2(R) + \cos^2(R) = 1

, substitute

\cos(R) = \frac{5}{13}

into the equation:

\sin^2(R) + \left(\frac{5}{13}\right)^2 = 1 \implies \sin^2(R) + \frac{25}{169} = 1 \implies \sin^2(R) = 1 - \frac{25}{169} = \frac{144}{169}

. Taking the positive square root because

R

is an acute angle gives

\sin(R) = \frac{12}{13}

.

This determines the sine of the angle from its known cosine value.

Key Concept

Trigonometric ratios of complementary angles and the Pythagorean identity in right triangles.

Question 82Question →

In triangle $PQR$ , point $S$ lies on side $PQ$ and point $T$ lies on side $PR$ such that segment $ST$ is parallel to segment $QR$ . The length of segment $PS$ is $x$ , the length of segment $SQ$ is $6$ , the length of segment $ST$ is $x + 2$ , and the length of segment $QR$ is $2x + 7$ . What is the length of segment $QR$ ?

Show answer & explanation

Answer: 15

Answer

The length of segment

QR

is

15

.

Since segment

ST

is parallel to segment

QR

, triangle

PST

is similar to triangle

PQR

. The ratio of their corresponding sides is equal, so

\frac{PS}{PQ} = \frac{ST}{QR}

. Substituting the values gives

\frac{x}{x + 6} = \frac{x + 2}{2x + 7}

. Cross-multiplying and simplifying yields

x^2 - x - 12 = 0

. Factoring gives

(x-4)(x+3)=0

, so

x=4

because side lengths must be positive. Substituting

x=4

into

2x+7

yields

15

.

Step-by-Step Solution

1

Determine the similarity between the triangles

Triangle

PST

is similar to triangle

PQR

Since segment

ST

is parallel to segment

QR

, the corresponding angles are equal, establishing similarity by Angle-Angle (AA) criterion.

2

Express the total length of side

PQ

PQ = x + 6

The length of side

PQ

is the sum of the collinear segments

PS

and

SQ

.

3

Set up the similarity ratio equation

\frac{x}{x + 6} = \frac{x + 2}{2x + 7}

Corresponding side lengths of similar triangles are in proportion:

\frac{PS}{PQ} = \frac{ST}{QR}

.

4

Solve the quadratic equation for

x

x = 4

Cross-multiplying gives

x(2x + 7) = (x + 2)(x + 6) \implies 2x^2 + 7x = x^2 + 8x + 12 \implies x^2 - x - 12 = 0 \implies (x - 4)(x + 3) = 0

. Since length must be positive,

x = 4

.

5

Calculate the length of segment

QR

QR = 15

Substitute

x = 4

into the expression for

QR

, which is

2x + 7

, resulting in

2(4) + 7 = 15

.

Key Concept

Using triangle similarity and algebraic equations to find unknown segment lengths.

Question 83Question →

For an acute angle $\theta$ , the expression $\frac{\sin^4(\theta) - \cos^4(\theta)}{\sin(\theta) - \cos(\theta)}$ is equal to $\frac{7}{5}$ . What is the value of $25\sin(\theta)\cos(\theta)$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12.

The expression

\frac{\sin^4(\theta) - \cos^4(\theta)}{\sin(\theta) - \cos(\theta)}

simplifies directly to

\sin(\theta) + \cos(\theta) = \frac{7}{5}

by applying the difference of squares identity twice and substituting the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

. Squaring both sides of this simplified equation gives

\sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{49}{25}

. Replacing

\sin^2(\theta) + \cos^2(\theta)

with 1 yields

1 + 2\sin(\theta)\cos(\theta) = \frac{49}{25}

, which simplifies to

\sin(\theta)\cos(\theta) = \frac{12}{25}

. Multiplying this result by 25 gives the final integer value of 12.

Step-by-Step Solution

1

Factor the numerator of the expression.

\sin^4(\theta) - \cos^4(\theta) = (\sin^2(\theta) - \cos^2(\theta))(\sin^2(\theta) + \cos^2(\theta))

The difference of squares identity can be applied to terms with fourth powers.

2

Apply the Pythagorean identity to simplify the factored numerator.

\sin^4(\theta) - \cos^4(\theta) = \sin^2(\theta) - \cos^2(\theta)

The Pythagorean trigonometric identity states that

\sin^2(\theta) + \cos^2(\theta) = 1

.

3

Factor the remaining term in the numerator.

\sin^2(\theta) - \cos^2(\theta) = (\sin(\theta) - \cos(\theta))(\sin(\theta) + \cos(\theta))

This is another application of the difference of squares identity.

4

Simplify the fraction by dividing the common factor in the numerator and denominator.

\sin(\theta) + \cos(\theta) = \frac{7}{5}

The term

\sin(\theta) - \cos(\theta)

in the numerator and denominator cancels out since

\theta

is an acute angle and

\sin(\theta) \neq \cos(\theta)

.

5

Square both sides of the simplified equation.

\sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{49}{25}

Squaring both sides allows us to relate the sum

\sin(\theta) + \cos(\theta)

to the product

\sin(\theta)\cos(\theta)

.

6

Substitute the Pythagorean identity and solve for the product of sine and cosine.

\sin(\theta)\cos(\theta) = \frac{12}{25}

Substituting

\sin^2(\theta) + \cos^2(\theta) = 1

yields

1 + 2\sin(\theta)\cos(\theta) = \frac{49}{25}

, which simplifies to

2\sin(\theta)\cos(\theta) = \frac{24}{25}

.

7

Multiply the product by 25 to find the target value.

12

The question asks for the value of

25\sin(\theta)\cos(\theta)

.

Key Concept

Simplifying trigonometric expressions using algebraic factorization and Pythagorean identities.

Alternative Method

Since

\theta

is an acute angle in a right triangle, we can test standard Pythagorean triples. A right triangle with side lengths 3, 4, and 5 has an angle

\theta

where

\sin(\theta) = \frac{3}{5}

and

\cos(\theta) = \frac{4}{5}

. Checking these values in the simplified expression gives

\sin(\theta) + \cos(\theta) = \frac{3}{5} + \frac{4}{5} = \frac{7}{5}

, which matches the given condition. We can then directly calculate

25\sin(\theta)\cos(\theta) = 25 \left(\frac{3}{5}\right)\left(\frac{4}{5}\right) = 12

.

Estimated Time:3m 0s

Question 84Question →

A circle has a central angle of $60^\circ$ that subtends an arc of length $4\pi$ . What is the circumference of the circle?

12\pi

16\pi

24\pi

240\pi

Show answer & explanation

Answer: $24\pi$

Answer

The circumference of the circle is

24\pi

.

The correct answer is

24\pi

. A full circle is

360^\circ

. A central angle of

60^\circ

represents

\frac{60^\circ}{360^\circ} = \frac{1}{6}

of the entire circle. Therefore, the length of the arc subtended by this angle is

\frac{1}{6}

of the total circumference. To find the circumference, we multiply the arc length by

6

:

4\pi \times 6 = 24\pi

.

Step-by-Step Solution

1

Find the fraction of the circle represented by the

60^\circ

central angle.

\frac{60^\circ}{360^\circ} = \frac{1}{6}

A full circle has a central angle of

360^\circ

, so the fraction of the circle is the ratio of the given central angle to

360^\circ

.

2

Set up the proportion relating the arc length to the total circumference (

C

).

\frac{1}{6} = \frac{4\pi}{C}

The ratio of the arc length to the total circumference equals the ratio of the central angle to the total degree measure of a circle.

3

Solve for the circumference (

C

) by multiplying both sides by

6

.

C = 6 \times 4\pi = 24\pi

Multiplying the arc length of the sector by the reciprocal of the fraction yields the total circumference of the circle.

Key Concept

Finding the circumference of a circle given the central angle and the subtended arc length using proportional reasoning.

Question 85Question →

In a right triangle, the two acute angles are $\theta$ and $\phi$ . If $\sin(\theta) = \frac{7}{25}$ and $\cos(\phi) = \frac{k}{50}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 14

Answer

14

In any right triangle, the two acute angles,

\theta

and

\phi

, must sum to

90^\circ

. The cofunction trigonometric identity states that

\sin(\theta) = \cos(90^\circ - \theta) = \cos(\phi)

. Therefore, we can equate the two given values:

\frac{7}{25} = \frac{k}{50}

. Solving this equation for

k

gives

k = \frac{7 \times 50}{25} = 14

.

Step-by-Step Solution

1

Determine the relationship between the two acute angles in a right triangle.

\theta + \phi = 90^\circ

Since the sum of the angles in any triangle is

180^\circ

and a right triangle has one

90^\circ

angle, the sum of the other two acute angles must be

90^\circ

.

2

Apply the cofunction trigonometric identity for complementary angles.

\sin(\theta) = \cos(\phi)

The sine of an acute angle is always equal to the cosine of its complement.

3

Equate the given expressions and solve for

k

.

\frac{7}{25} = \frac{k}{50} \implies k = 14

Substitute the given values into the identity and multiply both sides by

50

to isolate

k

.

Key Concept

Cofunction identities for complementary angles in a right triangle

Question 86Question →

In a circle with center $O$ , central angle $AOB$ has a measure of $\frac{5\pi}{6}$ radians. If the radius of the circle is $12$ , the length of arc $AB$ is $k\pi$ . What is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The arc length

s

subtended by a central angle

\theta

(in radians) in a circle of radius

r

is given by

s = r\theta

. Substituting the radius

r = 12

and the angle

\theta = \frac{5\pi}{6}

yields

s = 12 \times \frac{5\pi}{6} = 10\pi

. Since the arc length is expressed as

k\pi

, the value of

k

is

10

.

Step-by-Step Solution

1

Identify the formula for arc length in radians.

s = r\theta

The arc length

s

is directly proportional to the radius

r

and the central angle

\theta

in radians.

2

Substitute the given values into the formula.

s = 12 \times \frac{5\pi}{6}

The radius of the circle is

12

and the central angle is

\frac{5\pi}{6}

radians.

3

Calculate the arc length.

s = 10\pi

Simplifying

12 \times \frac{5}{6}

gives

2 \times 5 = 10

, so the product is

10\pi

.

4

Determine the value of

k

.

k = 10

We equate the calculated arc length

10\pi

with the given expression

k\pi

.

Key Concept

Arc length of a circle using radian measure

Estimated Time:45s

Question 87Question →

In the $xy$ -plane, the terminal ray of an angle $\theta$ in standard position intersects the unit circle at a point $P$ . If the line passing through $P$ and the point $(0, 1)$ has a slope of $-\frac{1}{2}$ , what is the value of $\sin(\theta)$ ?

\frac{4}{5}

-\frac{3}{5}

\frac{3}{5}

1

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

three-fifths

The correct answer is three-fifths. Since the point

P

lies on the unit circle, its coordinates are

(\cos(\theta), \sin(\theta))

. Using the slope formula between

P

and the point

(0, 1)

, we get the equation

\frac{\sin(\theta) - 1}{\cos(\theta)} = -\frac{1}{2}

. Cross-multiplying and rearranging gives

\cos(\theta) = 2(1 - \sin(\theta))

. Substituting this into the Pythagorean identity

\cos^2(\theta) + \sin^2(\theta) = 1

yields the quadratic equation

5\sin^2(\theta) - 8\sin(\theta) + 3 = 0

. Factoring this equation gives

(5\sin(\theta) - 3)(\sin(\theta) - 1) = 0

, which gives the solutions

\sin(\theta) = \frac{3}{5}

or

\sin(\theta) = 1

. Since the line is defined by two distinct points,

P

cannot be

(0, 1)

, which means

\sin(\theta) \neq 1

. Thus,

\sin(\theta) = \frac{3}{5}

.

Step-by-Step Solution

1

Represent the point

P

on the unit circle using trigonometric functions and set up the slope equation with the point

(0,1)

.

The slope equation is

\frac{\sin(\theta) - 1}{\cos(\theta)} = -\frac{1}{2}

.

Since

P

lies on the unit circle, its coordinates are

(\cos(\theta), \sin(\theta))

. The slope of a line through

(x_1, y_1)

and

(x_2, y_2)

is

\frac{y_2 - y_1}{x_2 - x_1}

.

2

Express

\cos(\theta)

in terms of

\sin(\theta)

by cross-multiplying and isolating

\cos(\theta)

in the equation.

\cos(\theta) = 2(1 - \sin(\theta))

.

This substitution variable will allow us to rewrite the Pythagorean identity in terms of a single variable,

\sin(\theta)

.

3

Substitute the expression for

\cos(\theta)

into the Pythagorean trigonometric identity

\cos^2(\theta) + \sin^2(\theta) = 1

and simplify.

4(1 - \sin(\theta))^2 + \sin^2(\theta) = 1 \implies 5\sin^2(\theta) - 8\sin(\theta) + 3 = 0

.

The Pythagorean identity is a fundamental relationship between the sine and cosine of any angle.

4

Factor the quadratic equation to find the possible values of

\sin(\theta)

, and reject any extraneous solutions.

(5\sin(\theta) - 3)(\sin(\theta) - 1) = 0

, giving

\sin(\theta) = \frac{3}{5}

or

\sin(\theta) = 1

. The value

\sin(\theta) = 1

is rejected because it makes the point

P

identical to

(0, 1)

, meaning a line cannot be defined.

A line requires two distinct points to be defined with a specific slope.

Key Concept

Unit circle definitions and the Pythagorean trigonometric identity

Question 88Question →

In the figure, lines $AB$ and $CD$ intersect at point $O$ , and ray $OE$ is perpendicular to line $AB$ . If the measure of $\angle EOC$ is $27^\circ$ , what is the measure, in degrees, of $\angle BOD$ ?

Show answer & explanation

Answer: 63

Answer

The measure of

\angle BOD

is 63 degrees.

The correct answer is 63. Since ray

OE

is perpendicular to line

AB

, the angle

\angle EOA

is a right angle measuring

90^\circ

. The adjacent angles

\angle AOC

and

\angle EOC

make up

\angle EOA

, which means

\angle AOC = 90^\circ - 27^\circ = 63^\circ

. Finally, because lines

AB

and

CD

intersect at point

O

, the angle

\angle BOD

and the angle

\angle AOC

are vertical angles. Vertical angles are equal in measure, so the measure of

\angle BOD

is

63^\circ

.

Step-by-Step Solution

1

Identify the angle formed by perpendicular lines.

The measure of

\angle EOA

is

90^\circ

.

Since ray

OE

is perpendicular to line

AB

, the angle

\angle EOA

is a right angle.

2

Calculate the measure of

\angle AOC

.

The measure of

\angle AOC

is

63^\circ

.

Angles

\angle AOC

and

\angle EOC

are adjacent and form the right angle

\angle EOA

, meaning they are complementary:

\angle AOC = 90^\circ - 27^\circ = 63^\circ

.

3

Determine the measure of

\angle BOD

.

The measure of

\angle BOD

is

63^\circ

.

Lines

AB

and

CD

intersect at point

O

, making

\angle BOD

and

\angle AOC

vertical angles. Since vertical angles are equal, the measure of

\angle BOD

is equal to the measure of

\angle AOC

.

Key Concept

Using properties of perpendicular lines and vertical angles to solve for unknown angle measures.

Estimated Time:1m 30s

Question 89Question →

In triangle $ABC$ , the measure of angle $ABC$ is $90^\circ$ and the measure of angle $BAC$ is $30^\circ$ . Segment $BD$ is perpendicular to segment $AC$ such that $D$ lies on $AC$ . Segment $DE$ is perpendicular to segment $BC$ such that $E$ lies on $BC$ . If the length of segment $CE$ is $3$ , what is the length of segment $AC$ ?

Show answer & explanation

Answer: 24

Answer

The correct answer is 24.

The correct answer is 24. Since the question asks for the length of segment AC, and by sequentially applying the properties of 30-60-90 right triangles we find CD = 6, BC = 12, and AC = 24.

Step-by-Step Solution

1

Find the measure of angle

ACB

in triangle

ABC

.

\angle ACB = 60^\circ

The sum of the angles in a triangle is

180^\circ

, so

\angle ACB = 180^\circ - \angle ABC - \angle BAC = 180^\circ - 90^\circ - 30^\circ = 60^\circ

.

2

Calculate the length of segment

CD

using right triangle

DEC

.

CD = 6

In the right triangle

DEC

,

\angle DEC = 90^\circ

and

\angle C = 60^\circ

, making it a 30-60-90 triangle. The side opposite the

30^\circ

angle is

CE = 3

, so the hypotenuse

CD

is

2 \times CE = 2 \times 3 = 6

.

3

Calculate the length of segment

BC

using right triangle

BDC

.

BC = 12

In the right triangle

BDC

,

BD

is perpendicular to

AC

, so

\angle BDC = 90^\circ

. With

\angle BCD = 60^\circ

, this is a 30-60-90 triangle. The side opposite the

30^\circ

angle is

CD = 6

, so the hypotenuse

BC

is

2 \times CD = 2 \times 6 = 12

.

4

Calculate the length of the hypotenuse

AC

using right triangle

ABC

.

AC = 24

In the right triangle

ABC

, the angle

\angle BAC = 30^\circ

and the side opposite to it is

BC = 12

. The hypotenuse

AC

is twice the length of the opposite leg, so

AC = 2 \times BC = 2 \times 12 = 24

.

Key Concept

Properties of special right triangles (30-60-90 triangles) and their trigonometric ratios.

Question 90Question →

In the $xy$ -plane, an angle $\theta$ in standard position has its terminal ray intersecting the unit circle at the point $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ . The angle is rotated counterclockwise by $15^\circ$ , and then its measure is doubled. The final resulting angle is coterminal with an angle of $\frac{a\pi}{b}$ radians, where $0 \le \frac{a\pi}{b} < 2\pi$ , and $a$ and $b$ are positive integers with no common factors. What is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. Starting with the point

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

on the unit circle, the angle in standard position is

225^\circ

(or

\frac{5\pi}{4}

radians). Adding

15^\circ

for the counterclockwise rotation yields

240^\circ

(or

\frac{4\pi}{3}

radians). Doubling this measure gives

480^\circ

(or

\frac{8\pi}{3}

radians). Finding the coterminal angle within

[0, 2\pi)

yields

120^\circ

(or

\frac{2\pi}{3}

radians). In this form,

a=2

and

b=3

, which are positive integers with no common factors, so

a+b = 5

.

Step-by-Step Solution

1

Determine the initial angle of the terminal ray intersecting the unit circle at

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

.

The initial angle

\theta

is

225^\circ

(or

\frac{5\pi}{4}

radians).

Since both the

x

- and

y

-coordinates are negative and equal, the angle lies in the third quadrant and forms a

45^\circ

reference angle with the negative

x

-axis, which corresponds to

225^\circ

.

2

Apply a counterclockwise rotation of

15^\circ

to the initial angle.

The new angle is

240^\circ

(or

\frac{4\pi}{3}

radians).

Counterclockwise rotations correspond to adding positive angle measures:

225^\circ + 15^\circ = 240^\circ

.

3

Double the measure of the rotated angle.

The doubled angle is

480^\circ

(or

\frac{8\pi}{3}

radians).

The prompt specifies that the angle's measure is doubled after the rotation:

240^\circ \times 2 = 480^\circ

.

4

Find the coterminal angle within the standard interval

[0, 2\pi)

radians (or

[0^\circ, 360^\circ)

).

The coterminal angle is

120^\circ

(or

\frac{2\pi}{3}

radians).

To bring the angle back into the interval

[0^\circ, 360^\circ)

, subtract

360^\circ

(one full rotation):

480^\circ - 360^\circ = 120^\circ

. In radians,

120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}

radians.

5

Identify

a

and

b

and calculate their sum.

a = 2

,

b = 3

, and

a + b = 5

.

The coterminal angle is expressed as

\frac{a\pi}{b}

in simplest form, so

a=2

and

b=3

, which are positive integers with no common factors.

Key Concept

Converting between radians and degrees, finding coterminal angles, and performing angle transformations.

Estimated Time:3m 0s

Question 91Question →

A circle has a circumference of $16\pi$ . An arc on this circle has a length of $2\pi$ . What is the measure, in degrees, of the central angle that subtends this arc?

22.5

40

45

90

Show answer & explanation

Answer: $45$

Answer

45

The correct answer is

45

because the ratio of the arc length to the total circumference is

\frac{2\pi}{16\pi} = \frac{1}{8}

. Since a full circle measures

360^\circ

, the central angle that subtends this arc is

\frac{1}{8} \times 360^\circ = 45^\circ

.

Step-by-Step Solution

1

Find the ratio of the arc length to the total circumference of the circle.

The ratio is

\frac{2\pi}{16\pi} = \frac{1}{8}

.

An arc length represents a fraction of the circle's total perimeter (circumference).

2

Multiply this fraction by the total degree measure of a circle (

360^\circ

) to find the measure of the central angle.

\frac{1}{8} \times 360^\circ = 45^\circ

.

The measure of the central angle is proportional to the fraction of the circle's circumference that the arc covers.

Key Concept

The relationship between arc length, circumference, and central angle measure in a circle.

Question 92Question →

An acute angle $\theta$ satisfies the equation $\sin(\theta) = \frac{24}{25}$ . What is the value of $\frac{1}{\cos(\theta)} - \tan(\theta)$ ?

\frac{3}{4}

\frac{1}{7}

7

\frac{1}{24}

Show answer & explanation

Answer: $\frac{1}{7}$

Answer

The correct value of the expression is

\frac{1}{7}

.

The correct answer is

\frac{1}{7}

. Using the Pythagorean identity, we determine that

\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \frac{7}{25}

. Using the definition of tangent, we find

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{24}{7}

. Substituting these values into the given expression yields

\frac{1}{7/25} - \frac{24}{7} = \frac{25}{7} - \frac{24}{7} = \frac{1}{7}

.

Step-by-Step Solution

1

Find the value of

\cos(\theta)

using the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

.

\cos(\theta) = \frac{7}{25}

Since

\theta

is an acute angle,

\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{24}{25}\right)^2} = \sqrt{\frac{49}{625}} = \frac{7}{25}

.

2

Find the value of

\tan(\theta)

using the definition

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

.

\tan(\theta) = \frac{24}{7}

Substituting the known values gives

\tan(\theta) = \frac{24/25}{7/25} = \frac{24}{7}

.

3

Substitute the values of

\cos(\theta)

and

\tan(\theta)

into the expression

\frac{1}{\cos(\theta)} - \tan(\theta)

and simplify.

\frac{1}{7}

The expression becomes

\frac{1}{7/25} - \frac{24}{7} = \frac{25}{7} - \frac{24}{7} = \frac{1}{7}

.

Key Concept

Pythagorean identity and trigonometric ratio definitions in a right triangle

Question 93Question →

In the $xy$ -plane, a circle with center $(4, 9)$ and radius $5$ is defined by the equation $(x - h)^2 + (y - 9)^2 = 25$ , where $h$ is a positive constant. What is the value of $h$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

h

is

4

.

The standard equation of a circle with center

(h, k)

and radius

r

is

(x - h)^2 + (y - k)^2 = r^2

. Since the center is

(4, 9)

and the radius is

5

, the equation of the circle is

(x - 4)^2 + (y - 9)^2 = 5^2

, which simplifies to

(x - 4)^2 + (y - 9)^2 = 25

. Comparing this to the given equation

(x - h)^2 + (y - 9)^2 = 25

, we see that the constant

h

corresponds to the

x

-coordinate of the center, which is

4

.

Step-by-Step Solution

1

Identify the standard equation of a circle.

(x - h)^2 + (y - k)^2 = r^2

This formula represents a circle with center

(h, k)

and radius

r

in the coordinate plane.

2

Substitute the given center coordinates into the standard equation.

For center

(4, 9)

, the equation is

(x - 4)^2 + (y - 9)^2 = r^2

.

The coordinates of the center are mapped directly to

h

and

k

in the standard form.

3

Compare the equation template to find the value of

h

.

The equation

(x - 4)^2 + (y - 9)^2 = 25

matches the format

(x - h)^2 + (y - 9)^2 = 25

, indicating that

h = 4

.

Equating the terms in both equations allows us to identify the value of the positive constant

h

.

Key Concept

Extracting coordinates of the center from the standard form equation of a circle.

Question 94Question →

For an acute angle $\theta$ , the expression $\frac{\sin(\theta) + \cos(\theta)}{\sec(\theta) + \csc(\theta)}$ is equal to $k$ . If $\tan(\theta) = 3$ , what is the value of $k$ ?

\frac{3}{5}

\frac{3}{10}

3

\frac{8}{5}

Show answer & explanation

Answer: $\frac{3}{10}$

Answer

The correct answer is three-tenths, or 3/10.

The correct answer is

\frac{3}{10}

. By rewriting the denominator using reciprocal identities, we have

\sec(\theta) + \csc(\theta) = \frac{1}{\cos(\theta)} + \frac{1}{\sin(\theta)} = \frac{\sin(\theta) + \cos(\theta)}{\sin(\theta)\cos(\theta)}

. Dividing the numerator by this expression simplifies the entire fraction to

\sin(\theta)\cos(\theta)

. Using the reference right triangle for

\tan(\theta) = 3

, we find that

\sin(\theta) = \frac{3}{\sqrt{10}}

and

\cos(\theta) = \frac{1}{\sqrt{10}}

, which gives a product of

\frac{3}{10}

.

Step-by-Step Solution

1

Rewrite the terms in the denominator using their reciprocal definitions.

\sec(\theta) = \frac{1}{\cos(\theta)}

and

\csc(\theta) = \frac{1}{\sin(\theta)}

, so the expression becomes

\frac{\sin(\theta) + \cos(\theta)}{\frac{1}{\cos(\theta)} + \frac{1}{\sin(\theta)}}

.

Expressing secant and cosecant in terms of sine and cosine allows for the simplification of the complex fraction.

2

Combine the terms in the denominator by finding a common denominator.

\frac{1}{\cos(\theta)} + \frac{1}{\sin(\theta)} = \frac{\sin(\theta) + \cos(\theta)}{\sin(\theta)\cos(\theta)}

.

Finding a common denominator simplifies the addition of the fractional terms.

3

Divide the numerator by the simplified denominator.

\frac{\sin(\theta) + \cos(\theta)}{\frac{\sin(\theta) + \cos(\theta)}{\sin(\theta)\cos(\theta)}} = \sin(\theta)\cos(\theta)

.

Since

\theta

is an acute angle, the sum

\sin(\theta) + \cos(\theta)

is non-zero, allowing it to be canceled from both the numerator and the denominator.

4

Find the values of

\sin(\theta)

and

\cos(\theta)

using the given relation

\tan(\theta) = 3

.

Construct a right triangle where the opposite side to angle

\theta

is

3

and the adjacent side is

1

. The hypotenuse is

\sqrt{3^2 + 1^2} = \sqrt{10}

. Therefore,

\sin(\theta) = \frac{3}{\sqrt{10}}

and

\cos(\theta) = \frac{1}{\sqrt{10}}

.

Using a reference right triangle is a direct way to find the values of other trigonometric functions from a known tangent value.

5

Multiply the values of

\sin(\theta)

and

\cos(\theta)

to find the value of

k

.

k = \sin(\theta)\cos(\theta) = \left(\frac{3}{\sqrt{10}}\right)\left(\frac{1}{\sqrt{10}}\right) = \frac{3}{10}

.

This yields the final numerical value of the expression.

Key Concept

Simplification of trigonometric expressions using reciprocal identities and finding trigonometric ratios from a given tangent ratio.

Question 95Question →

If $\sin(2x + 10)^\circ = \cos(3x - 5)^\circ$ , where the measures of both angles are in degrees and are acute, what is the value of $x$ ?

Show answer & explanation

Answer: 17

Answer

17

According to the co-function identities of trigonometry, the sine of an acute angle is equal to the cosine of its complementary angle. Therefore, if

\sin(2x + 10)^\circ = \cos(3x - 5)^\circ

, the sum of the two angles must be

90^\circ

. This gives the equation

(2x + 10) + (3x - 5) = 90

. Simplifying the equation yields

5x + 5 = 90

. Subtracting 5 from both sides gives

5x = 85

, and dividing by 5 yields

x = 17

.

Step-by-Step Solution

1

Relate the sine and cosine functions using the co-function identity.

Since

\sin(A) = \cos(B)

for acute angles, the angles must be complementary, so

A + B = 90^\circ

.

The co-function identity states that the sine of an angle is equal to the cosine of its complement.

2

Set up the algebraic equation using the given angle expressions.

(2x + 10) + (3x - 5) = 90

This expresses the condition that the sum of the two acute angles is equal to

90^\circ

.

3

Solve the equation for

x

.

5x + 5 = 90 \implies 5x = 85 \implies x = 17

Combine like terms and isolate

x

by performing basic arithmetic operations.

Key Concept

Co-function identities relate trigonometric functions of complementary angles, specifically

\sin(\theta) = \cos(90^\circ - \theta)

.

Question 96Question →

In the diagram shown, point $D$ lies on side $AB$ of triangle $ABC$ , and point $E$ lies on side $AC$ . The lengths of the segments are $AB = 20$ , $AC = 16$ , $AD = 8$ , and $AE = 10$ . If the area of quadrilateral $BCED$ is $54$ , what is the area of triangle $ADE$ ?

(Note: Figure not drawn to scale.)

Show answer & explanation

Answer: 18

Answer

18

Triangles ADE and ACB share angle A. The ratios of the adjacent sides of angle A are AD/AC = 8/16 = 1/2 and AE/AB = 10/20 = 1/2. By the Side-Angle-Side (SAS) similarity theorem, triangle ADE is similar to triangle ACB. The ratio of their areas is equal to the square of their similarity ratio: (1/2)^2 = 1/4. Therefore, the area of triangle ACB is 4 times the area of triangle ADE. The area of quadrilateral BCED is the difference between the area of triangle ACB and the area of triangle ADE, which is 4 * Area(ADE) - Area(ADE) = 3 * Area(ADE). Since the area of quadrilateral BCED is 54, we have 3 * Area(ADE) = 54, which simplifies to Area(ADE) = 18.

Step-by-Step Solution

1

Calculate side ratios to establish similarity.

The ratio of AD to AC is 8/16 = 1/2, and the ratio of AE to AB is 10/20 = 1/2.

Checking if the corresponding sides surrounding the shared angle are in the same proportion.

2

Apply the Side-Angle-Side (SAS) similarity theorem.

Triangle ADE is similar to triangle ACB (triangle ADE ~ triangle ACB), where vertex A corresponds to A, D corresponds to C, and E corresponds to B.

Since the ratio of two pairs of corresponding sides is equal and their included angle is congruent, the triangles are similar.

3

Determine the area ratio based on the similarity scale factor.

The ratio of the area of triangle ADE to the area of triangle ACB is (1/2)^2 = 1/4.

The ratio of the areas of two similar figures is equal to the square of their similarity ratio.

4

Relate the area of the quadrilateral to the area of the smaller triangle.

Area(BCED) = Area(ACB) - Area(ADE) = 4 * Area(ADE) - Area(ADE) = 3 * Area(ADE).

The area of the quadrilateral is the difference between the areas of the larger and smaller triangles.

5

Solve for the area of triangle ADE.

Area(ADE) = 54 / 3 = 18.

Dividing the given area of the quadrilateral by 3 yields the area of the smaller triangle.

Key Concept

SAS Triangle Similarity and the Area Ratios of Similar Triangles

Estimated Time:2m 30s

Question 97Question →

In the $xy$ -plane, an angle $\theta$ is in standard position. The terminal ray of $\theta$ is rotated counterclockwise by $\frac{7\pi}{6}$ radians, and then rotated clockwise by $135^\circ$ . If the terminal ray of the resulting angle lies on the line $y = -x$ in the fourth quadrant, and the original angle $\theta$ has a measure of $d$ degrees, where $0 \leq d < 360$ , what is the value of $d$ ?

Show answer & explanation

Answer: 240

Answer

The value of

d

is 240.

A terminal ray in the fourth quadrant lying on the line

y = -x

forms a

315^\circ

angle in standard position. The first rotation is counterclockwise by

\frac{7\pi}{6}

radians, which is equivalent to

\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ

. The second rotation is clockwise by

135^\circ

, representing a decrease of

135^\circ

. Therefore, the total transformation is

\theta + 210^\circ - 135^\circ = 315^\circ

. Solving for

\theta

gives

\theta + 75^\circ = 315^\circ

, which simplifies to

\theta = 240^\circ

. Since

240^\circ

lies in the interval

[0, 360)

, the value of

d

is 240.

Step-by-Step Solution

1

Determine the angle in standard position for a terminal ray on the line

y = -x

in the fourth quadrant.

The terminal ray corresponds to an angle of

315^\circ

(or any angle coterminal with it).

The line

y = -x

in the fourth quadrant makes an angle of

45^\circ

below the positive

x

-axis, which corresponds to

360^\circ - 45^\circ = 315^\circ

in standard position.

2

Convert the counterclockwise rotation from radians to degrees.

\frac{7\pi}{6} \text{ radians} = 210^\circ

.

To convert radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

.

3

Express the rotations mathematically and set up the equation for

\theta

.

\theta + 210^\circ - 135^\circ = 315^\circ + 360^\circ n

(where

n

is an integer).

In standard position, counterclockwise rotations represent positive changes in angle measure, whereas clockwise rotations represent negative changes in angle measure.

4

Solve for

\theta

and apply the domain restriction

0 \leq d < 360

.

\theta = 240^\circ

, so

d = 240

.

Simplifying the equation gives

\theta + 75^\circ = 315^\circ

, which yields

\theta = 240^\circ

when

n=0

.

Key Concept

Converting angles from radians to degrees, understanding the direction of rotation, and determining standard position angles on the coordinate plane.

Question 98Question →

The equation of a circle in the $xy$ -plane is $(x - 8)^2 + (y + 1)^2 = 81$ . What are the coordinates of the center of the circle?

(-8, 1)

(8, 1)

(8, -1)

(-8, -1)

Show answer & explanation

Answer: $(8, -1)$

Answer

The center of the circle is

(8, -1)

The standard form of the equation of a circle in the

xy

-plane is

(x - h)^2 + (y - k)^2 = r^2

, where the center is

(h, k)

. In the given equation

(x - 8)^2 + (y + 1)^2 = 81

, comparing the terms to the standard form gives

h = 8

and

k = -1

. Thus, the center of the circle is

(8, -1)

.

Step-by-Step Solution

1

State the standard form of a circle's equation

The standard equation of a circle with center

(h, k)

and radius

r

is

(x - h)^2 + (y - k)^2 = r^2

.

This establishes the template to match the given equation against.

2

Rewrite the given equation to match the standard form exactly

The given equation

(x - 8)^2 + (y + 1)^2 = 81

can be rewritten as

(x - 8)^2 + (y - (-1))^2 = 9^2

.

Rewriting helps clearly identify the signs of

h

and

k

since the standard form uses subtraction:

(x - h)

and

(y - k)

.

3

Extract the center coordinates

(h, k)

h = 8

and

k = -1

, giving the coordinates

(8, -1)

.

By matching the rewritten equation to the standard form, we find the specific values of

h

and

k

that define the circle's center.

Key Concept

Identifying the center of a circle from its standard equation form

Estimated Time:45s

Question 99Question →

In right triangle $DEF$ , the measure of angle $F$ is $90^\circ$ . If $\cos(D) = \frac{3}{5}$ , what is the value of $\sin(E) - \tan(D)$ ?

-\frac{8}{15}

-\frac{3}{20}

-\frac{11}{15}

\frac{29}{15}

Show answer & explanation

Answer: $-\frac{11}{15}$

Answer

-\frac{11}{15}

To find the value of

\sin(E) - \tan(D)

, we first determine the trigonometric ratios for triangle

DEF

. Since

D

and

E

are the two acute angles in right triangle

DEF

with the right angle at

F

, the angles are complementary, which means

\sin(E) = \cos(D) = \frac{3}{5}

. Next, using the definition of cosine,

\cos(D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}

. By the Pythagorean theorem, the length of the opposite side to angle

D

is

\sqrt{5^2 - 3^2} = 4

. Therefore,

\tan(D) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}

. Substituting these values into the expression gives

\sin(E) - \tan(D) = \frac{3}{5} - \frac{4}{3} = \frac{9 - 20}{15} = -\frac{11}{15}

.

Step-by-Step Solution

1

Identify the relationship between the acute angles in right triangle

DEF

.

Since angle

F

is

90^\circ

, angles

D

and

E

are complementary, meaning

D + E = 90^\circ

. By the co-function identity,

\sin(E) = \cos(D)

.

This allows us to determine

\sin(E)

directly from the given value of

\cos(D)

without finding angle measures.

2

Determine the value of

\sin(E)

and find the side lengths of triangle

DEF

.

\sin(E) = \cos(D) = \frac{3}{5}

. Using the ratio

\cos(D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}

, we let the adjacent side

DF = 3

and the hypotenuse

DE = 5

. By the Pythagorean theorem, the opposite side is

EF = \sqrt{5^2 - 3^2} = 4

.

These side lengths are necessary to calculate the value of

\tan(D)

.

3

Calculate the value of

\tan(D)

and evaluate the given expression.

\tan(D) = \frac{\text{opposite}}{\text{adjacent}} = \frac{EF}{DF} = \frac{4}{3}

. Evaluating the expression:

\sin(E) - \tan(D) = \frac{3}{5} - \frac{4}{3} = \frac{9 - 20}{15} = -\frac{11}{15}

.

This completes the subtraction to find the final value of the expression.

Key Concept

Trigonometric ratios in right triangles and co-function identities of complementary angles.

Alternative Method

Instead of using the co-function identity, one can draw a right triangle

DEF

with adjacent side

DF = 3

and hypotenuse

DE = 5

. Using the Pythagorean theorem, the opposite side

EF = 4

. From the triangle,

\sin(E) = \frac{DF}{DE} = \frac{3}{5}

and

\tan(D) = \frac{EF}{DF} = \frac{4}{3}

. Evaluating the expression gives

\frac{3}{5} - \frac{4}{3} = -\frac{11}{15}

.

Estimated Time:1m 30s

Question 100Question →

In the $xy$ -plane, the graph of the equation $x^2 - 8x + y^2 = 0$ is a circle. What is the radius of the circle?

Show answer & explanation

Answer: 4

Answer

The radius of the circle is 4.

By completing the square on the

x

-terms in the equation

x^2 - 8x + y^2 = 0

, we add

( -8/2 )^2 = 16

to both sides, yielding

(x - 4)^2 + y^2 = 16

. Comparing this to the standard equation of a circle,

(x-h)^2 + (y-k)^2 = r^2

, we find that

r^2 = 16

. Taking the square root of both sides gives a radius of

4

.

Step-by-Step Solution

1

Group the

x

-terms together and prepare to complete the square.

(x^2 - 8x) + y^2 = 0

To write the equation in the standard form of a circle,

(x-h)^2 + (y-k)^2 = r^2

, we need to complete the square for the quadratic expression in

x

.

2

Complete the square for the

x

terms by adding

16

to both sides of the equation.

(x^2 - 8x + 16) + y^2 = 16

, which simplifies to

(x - 4)^2 + y^2 = 16

.

Adding

( -8/2 )^2 = 16

to both sides creates a perfect square trinomial

(x-4)^2

on the left side of the equation.

3

Identify the value of

r^2

and find the radius

r

.

r^2 = 16

, which gives

r = \sqrt{16} = 4

.

Comparing the equation

(x - 4)^2 + y^2 = 16

to the standard form

(x-h)^2 + (y-k)^2 = r^2

shows that

r^2 = 16

. Since the radius must be positive,

r = 4

.

Key Concept

Completing the square to find the standard form equation of a circle,

(x-h)^2 + (y-k)^2 = r^2

, to determine its radius.

Geometry and Trigonometry

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer