Video Transcript
Which of the following is equivalent to sin of 23 degrees? Is it (A) cos of 67 degrees, (B) cos of 23 degrees, option (C) negative cos of 23
degrees, (D) one over cos of 23 degrees, or (E) cos of 157 degrees?
In order to answer this question, we will need to remind ourselves of the cofunction
trigonometric identities. These are really useful as we know that the trigonometric functions, sine, cosine,
and tangent, appear often within the context of right-angled triangles. In particular, we can draw the following triangle in which the sin of the angle 𝛼 is
equal to the ratio of the opposite side, let’s call that 𝑥, and the hypotenuse. We’ll call that 𝑦. Then, we know that the unlabeled angle is found by subtracting 90 degrees and 𝛼
degrees from 180. So it’s 90 minus 𝛼 degrees.
If we want to find a trigonometric identity that now links 𝑥, 𝑦, and the angle we
just labeled, we need to use the cosine function. It’s cos of 𝜃 equals adjacent over hypotenuse. And we use 𝜃 for a moment to express the most general case. In this case, that’s cos of 90 minus 𝛼 equals 𝑥 over 𝑦.
We now see that both sin 𝛼 and cos of 90 minus 𝛼 are equal to 𝑥 over 𝑦. So they must also be equal to one another. Note at this stage that we could do this much more stringently using the unit
circle. But for our purposes, this is a nice demonstration of the cofunction identity we
need. That is, sin of 𝛼 equals cos of 90 minus 𝛼. Looking at the expression in the question, we need to let 𝛼 equal 23. Then, we can say that sin of 23 degrees must be equal to cos of 90 minus 23 or cos of
67 degrees.
So we have our answer. The correct answer is option (A). sin of 23 degrees is equal to cos of 67 degrees.
