In right triangle XYZ, the measure of angle Y is 90^\circ. If \cos(X) + \sin(Z) = 1.6, wha… — SAT

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ . If $\cos(X) + \sin(Z) = 1.6$ , what is the value of $\sin(X)$ ?

Answer: 0.6

0.6

The two acute angles in a right triangle sum to

90^\circ

, which means they are complementary. Therefore, the co-function identity

\sin(Z) = \cos(X)

applies. Substituting this relationship into the equation

\cos(X) + \sin(Z) = 1.6

gives

2\cos(X) = 1.6

, leading to

\cos(X) = 0.8

. Finally, using the Pythagorean identity

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

, we find

\sin(X) = \sqrt{1 - 0.8^2} = 0.6

Determine the relationship between angles

X

and

Z

X + Z = 90^\circ

Since the sum of the angles in right triangle

XYZ

180^\circ

and angle

Y

90^\circ

, the two acute angles

X

and

Z

must sum to

90^\circ

, making them complementary.

Use the co-function identity to relate

\sin(Z)

and

\cos(X)

\sin(Z) = \cos(X)

For complementary angles, the sine of one angle is equal to the cosine of the other.

Substitute the identity into the given equation to solve for

\cos(X)

\cos(X) = 0.8

Replacing

\sin(Z)

with

\cos(X)

\cos(X) + \sin(Z) = 1.6

yields

2\cos(X) = 1.6

, so

\cos(X) = 0.8

Solve for

\sin(X)

using the Pythagorean identity.

\sin(X) = 0.6

Since

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

, we have

\sin^2(X) + (0.8)^2 = 1

, which gives

\sin^2(X) = 1 - 0.64 = 0.36

. Since

X

is an acute angle, its sine must be positive, so

\sin(X) = 0.6

Co-function and Pythagorean trigonometric identities in right triangles

Question