Video Transcript
Find the value of cos of 90 degrees plus 𝜃 given sin 𝜃 is equal to three-fifths where 𝜃 is greater than zero degrees and less than 90 degrees.
We could solve this problem using the compound angle or addition angle formulae. However, in this case, we will use the co-function identities. Two of these identities state that sin of 90 degrees minus 𝜃 is equal to cos 𝜃, and cos of 90 degrees minus 𝜃 is equal to sin 𝜃. We can begin this question by rewriting 90 degrees as 180 degrees minus 90 degrees. Our expression becomes cos of 180 degrees minus 90 degrees plus 𝜃. This, in turn, can be rewritten as cos of 180 degrees minus 90 degrees minus 𝜃. We have, in effect, factored or factorized out negative one from the last two terms.
We also know from our CAST diagram that the sin of 180 degrees minus 𝜃 is equal to sin 𝜃 and the cos of 180 degrees minus 𝜃 is equal to negative cos 𝜃. Using the second one of these shows that cos of 180 degrees minus 90 degrees minus 𝜃 is equal to negative cos of 90 degrees minus 𝜃. As cos of 90 degrees minus 𝜃 is equal to sin 𝜃, negative cos of 90 degrees minus 𝜃 must be equal to negative sin 𝜃. In the question, we are told that sin 𝜃 is equal to three-fifths. Therefore, negative sin 𝜃 is equal to negative three-fifths. The value of cos 90 degrees plus 𝜃 is therefore equal to negative three-fifths.
