Solving Trig Identities Practice Problems

For the \(\displaystyle \sin \left( \right)\) and \(\displaystyle \cos \left( \right)\), they both contain a cos, and the cos half angle has the plus sign inside the radical.The tricky thing is that the sign of the whole identity depends on the quadrant where the angle \(\displaystyle \boldsymbol\) is (not angle \(A\), but \(\displaystyle \frac\)).

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Dividing In this problem, it is easier to start from the RHS. We multiply numerator and denominator of the fraction by the conjugate of the denominator.

\(\displaystyle \sin \theta =\frac\,\,\,\,\,\,\,\,\,\,\,\csc \theta =\frac\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cos \theta =\frac\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sec \theta =\frac\,\) \(\displaystyle \tan \theta =\frac=\frac\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cot \theta =\frac=\frac\) \(\begin\sin \left( \right)=\sin A\cos B \cos A\sin B\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cos \left( \right)=\cos A\cos B-\sin A\sin B\\\sin \left( \right)=\sin A\cos B-\cos A\sin B\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cos \left( \right)=\cos A\cos B \sin A\sin B\end\) \(\displaystyle \theta \theta =1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\theta 1=\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\theta 1=\theta \) An “identity” is something that is always true, so you are typically either substituting or trying to get two sides of an equation to equal each other.

We now proceed to derive two other related formulas that can be used when proving trigonometric identities.

It is suggested that you remember how to find the identities, rather than try to memorise each one.

Here are some examples of simple identity proofs with reciprocal and quotient identities.

Typically, to do these proofs, you must always start with one side (either side, but usually take the more complicated side) and manipulate the side until you end up with the other side.

Trig identities are sort of like puzzles since you have to “play” with them to get what you want.

You will also have to do some memorizing for these, since most of them aren’t really obvious.

Sometimes we have to find common denominators, like in the last example.

We didn’t need to turn it into sin and cos, since we only had tan and cot in the identity (although it still would have worked).

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