If \sin(2x + 10)^\circ = \cos(3x - 5)^\circ, where the measures of both angles are in degr… — SAT

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 $x$ ?

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

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.

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

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.

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

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

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