If \sin(x^\circ) = \cos(38^\circ), where 0 < x < 90, what is the value of x? — SAT

If $\sin(x^\circ) = \cos(38^\circ)$ , where $0 < x < 90$ , what is the value of $x$ ?

Answer: 52

The value of

x

is 52.

Applying the co-function identity

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

allows us to set the argument of the sine function as the complement of the cosine's argument. Thus,

x = 90 - 38 = 52

Identify the relevant trigonometric identity.

The co-function identity

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

is appropriate here.

We need to relate the sine of one angle to the cosine of another angle.

Equate the angle arguments using the identity.

\sin(x^\circ) = \cos(90^\circ - x^\circ) = \cos(38^\circ)

, which implies

90 - x = 38

For acute angles, if their cosine values are equal, the angles themselves must be equal.

Solve the linear equation for

x

x = 90 - 38 = 52

Isolating

x

gives the final answer.

Co-function identities relate the sine of an angle to the cosine of its complement:

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

Question