In a right triangle, the two acute angles are \theta and \phi. If \sin(\theta) = \frac{7}{… — SAT

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 $k$ ?

Answer: 14

Answer

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

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

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.

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

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