Question Video: Simplifying Trigonometric Expressions Using Cogunction Identities Mathematics

Simplify sin 𝜃 + cos (270° + 𝜃).

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Video Transcript

Simplify sin 𝜃 plus the cos of 270 degrees plus 𝜃.

In order to simplify this expression, we will begin by finding an equivalent expression to the cos of 270 degrees plus 𝜃. One way of doing this is by considering the unit circle. Adding 270 degrees to 𝜃 puts us in the same position on the unit circle as subtracting 90 degrees from 𝜃. The cos of 270 degrees plus 𝜃 is equal to the cos of 𝜃 minus 90 degrees. This is similar to one of our cofunction identities, which states that the cos of 90 degrees minus 𝜃 is equal to sin 𝜃. If we factor negative one out of the expression in our parentheses, we have the cos of negative 90 degrees minus 𝜃.

As cosine is an even function, the cos of negative 𝛼 is the same as the cos of 𝛼. This means that our expression is the same as the cos of 90 degrees minus 𝜃. We can then use the cofunction identity so that this is equal to sin 𝜃. Replacing the cos of 270 degrees plus 𝜃 with sin 𝜃, our original expression becomes sin 𝜃 plus sin 𝜃. This is equal to two sin 𝜃.

An alternative method would have been to have used the sum identities or addition formulae. One of these states that the cos of 𝛼 plus 𝛽 is equal to cos 𝛼 cos 𝛽 minus sin 𝛼 sin 𝛽. Considering our expression, we will let 𝛼 be 270 degrees and 𝛽 be 𝜃. This gives us cos of 270 degrees multiplied by cos 𝜃 minus sin of 270 degrees multiplied by sin 𝜃. The cos of 270 degrees is zero, and the sin of 270 degrees is negative one. Our expression simplifies to zero multiplied by cos 𝜃 minus negative one multiplied by sin 𝜃, which is equal to sin 𝜃. This confirms the expression we got using our first method. sin 𝜃 plus the cos of 270 degrees plus 𝜃 is equal to two sin 𝜃.

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