Video: The Relationship between Trigonometric Functions of Complementary Angles

Which of the following is equivalent to cos 40°? [A] − sin 40° [B] sin 40° [C] 1/sin 40° [D] sin 50° [E] sin 140°

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Video Transcript

Which of the following is equivalent to cos of 40 degrees? Is it A) negative sin of 40 degrees, B) sin of 40 degrees, C) one over sin of 40 degrees, D) sin of 50 degrees, or E) sin of 140 degrees.

We have been given an expression in terms of cos and five solutions in terms of sin. So let’s recall what we know about the relationship between the cosine and sine functions. There are two that spring to mind. For an angle 𝑥 degrees, sin of 90 minus 𝑥 is equal to cos of 𝑥. And cos of 90 minus 𝑥 is equal to sin of 𝑥. These are two correlated angle identities that we need to know by heart. So how did this help us and which one do we choose?

Well, we’ve been given cos of 40 degrees. And there’re two ways we could go about this. We could say that that’s the same as cos of 90 minus 50 degrees since 90 minus 50 is 40. And then, we use the second identity. This says that cos of 90 minus 𝑥 is the same as sin of 𝑥. So this means that cos of 90 minus 50 is the same as sin of 50 degrees. And that’s D. Cos of 40 degrees is equal to sin of 50 degrees.

Now actually, we could’ve used the first identity too. This way says that sin of 90 minus 𝑥 is equal to cos of 𝑥. So sin of 90 minus 40 is just equal to cos of 40.

And we then see that cos of 40 degrees once again is equal to sin of 50 degrees.

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