# Question Video: Using Relationship between the Sines and Cosines of Complementary Angles to Find the Value of a Trigonometric Function Mathematics

Find csc 𝛽 given 𝛼 and 𝛽 are two complementary angles, where sec 𝛼 = 5/4.

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### Video Transcript

Find csc 𝛽 given 𝛼 and 𝛽 are two complementary angles, where sec 𝛼 is equal to five-quarters.

We begin by recalling that two complementary angles sum to 90 degrees. In this question, this means that 𝛼 and 𝛽 sum to 90 degrees. Subtracting 𝛼 from both sides of this equation, we have 𝛽 is equal to 90 degrees minus 𝛼. We are trying to calculate csc of 𝛽. Therefore, this must be equal to the csc of 90 degrees minus 𝛼. As we are told in the question that the sec of 𝛼 is five-quarters, we’ll need to rewrite our expression using the reciprocal and cofunction identities.

One of the cofunction identities states that the sin of 90 degrees minus 𝜃 is equal to the cos of 𝜃. Since the cosecant and secant functions are the reciprocal of the sine and cosine functions, respectively, we have csc 𝜃 is equal to one over sin 𝜃 and sec 𝜃 is one over cos 𝜃. This means that the csc of 90 degrees minus 𝜃 is equal to the sec of 𝜃.

In this question, we have the csc of 90 degrees minus 𝛼, which is equal to sec 𝛼. And as already mentioned, we are told that this is equal to five-quarters. If 𝛼 and 𝛽 are two complementary angles where the sec of 𝛼 is five-quarters, then the csc of 𝛽 must also equal five-quarters.