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Section Trigonometry

We state here only the very minimal definitions and identities from trigonometry that we expect you to know. By “know” I mean memorize and be prepared to use on a test.

Definition 14.35.

The unit circle is the circle of radius one, centered at the origin.

Definition 14.36.

If \(P=(x,y)\) is a point on the unit circle and \(L\) is the line through the origin and \(P\text{,}\) and \(\theta\) is the angle between the \(x-\)axis and \(L\text{,}\) then we define cos\((\theta)\) to be \(x\) and sin\((\theta)\) to be \(y\text{.}\)

Definition 14.37.

These are the definitions of the remaining four trigonometric functions:

\begin{equation*} \tan(\theta )=\frac{\sin(\theta )}{\cos(\theta )}, \ \ \ \cot(\theta )=\frac{\cos(\theta )}{\sin(\theta )}, \ \ \ \csc(\theta )=\frac{1}{\sin(\theta )}, \ \ \ \mbox{ and } \ \ \ \sec(\theta )=\frac{1}{\cos(\theta)} \end{equation*}
Problem 14.38.

Graph each of \(\dsp \sin(-\theta )\text{,}\) \(\dsp -\sin(\theta )\text{,}\) \(\dsp \cos(-\theta )\) and \(\dsp \cos(\theta )\) to convince yourself of the following theorem.

The Pythagorean Identity follows immediately from Definition 14.36.

Problem 14.42.

Divide both sides of the Pythagorean Identity by \(\sin^2(\theta)\) to show that \(\dsp 1+\tan^{2}(\theta )=\sec^{2}(\theta )\text{.}\) Divide by \(\cos^2(\theta)\) to show that \(\dsp 1+\cot^{2}(\theta )=\csc^{2}(\theta )\text{.}\)

These next ones are a bit tricky to derive, but easy to remember if you memorize the Double Angle Identities.

Problem 14.45.

Use the first Sum/Difference Identity to prove the first Double Angle Identity.

Problem 14.46.

Use the second Sum/Difference Identity to prove the second Double Angle Identity.

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