In a right triangle, one of the acute angles is \theta. If \tan(\theta) = \frac{x}{y}, whe… — SAT

In a right triangle, one of the acute angles is $\theta$ . If $\tan(\theta) = \frac{x}{y}$ , where $x$ and $y$ are positive constants, what is the value of $\cos\left(\frac{\pi}{2} - \theta\right)$ in terms of $x$ and $y$ ?

A
$\frac{y}{\sqrt{x^2 + y^2}}$
$\frac{x}{\sqrt{x^2 + y^2}}$ Answer
C
$\frac{y}{x}$
D
$\frac{x}{x^2 + y^2}$

Answer

\frac{x}{\sqrt{x^2 + y^2}}

Using the cofunction identity,

\cos\left(\frac{\pi}{2} - \theta\right)

is equivalent to

\sin(\theta)

. For an angle

\theta

in a right triangle,

\tan(\theta) = \frac{x}{y}

defines the ratio of the opposite side (

x

) to the adjacent side (

y

). Applying the Pythagorean theorem, the hypotenuse is

\sqrt{x^2 + y^2}

. Therefore,

\sin(\theta)

, which is the ratio of the opposite side to the hypotenuse, is equal to

\frac{x}{\sqrt{x^2 + y^2}}

Step-by-Step Solution

Apply the cofunction identity to rewrite the target expression.

\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)

By the cofunction identities, the sine of an acute angle is equal to the cosine of its complementary angle.

Relate the given tangent ratio to the side lengths of a right triangle.

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{y}

, so the opposite side can be represented as

x

and the adjacent side as

y

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Apply the Pythagorean theorem to calculate the hypotenuse of the triangle.

\text{Hypotenuse} = \sqrt{x^2 + y^2}

The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Express the sine of the angle as a ratio of the opposite side to the hypotenuse.

\sin(\theta) = \frac{x}{\sqrt{x^2 + y^2}}

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

Key Concept

Cofunction identities and right triangle trigonometric ratios

Estimated Time:1m 30s

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