3.4 - Introduction to Trigonometric Identities
Contents
Objectives:
- Prove new trigonometric identities using already-proven identities and algebraic manipulation.
- Understand that there may be multiple ways of proving an identity.
- Solve trigonometric equations using identities as a tool.
- Definitions
- identity
Lesson Guide
Strategies for Proving Trigonometric Identities
A \underline{(trigonometric) identity} is an trigonometric equation which holds true for \emph{every} possible value of any variables involved.
Note: for our purposes in this course, we will rule out any values of a variable which result in division by zero.
Give several examples of trigonometric identities, and how we might show that they hold true for every possible value of the variable. Some possibilities are given below.
The equation $\cot(x)=\cos(x)\csc(x)$ represents an identity, since $\cos(x)\csc(x)=\cos(x)\cdot \frac{1}{\sin(x)}=
\frac{\cos(x)}{\sin(x)}=\cot(x)$.
- In order to show the above equation represented an identity, we rewrote one side of the equation ($\cos(x)\csc(x)$) so that it matched the other side ($\cot(x)$).
The equation $\cos^2(x)=1-\sin^2(x)$ represents an identity, since we can add $\sin^2(x)$ to both sides in order to obtain $\cos^2(x)+\sin^2(x)=1$, which is known to be a true statement for any value of $x$ (and in fact is the Pythagorean identity).
- Our method for establishing this identity is distinct from the first example. Here, we algebraically manipulated the desired equation to reduce it to a previously-proved identity. This is an excellent strategy for showing that an equation is an identity, and we'll use it frequently.
- What we are really saying here is that we can \emph{start} with a known identity (the Pythagorean identity), and subtract $\sin^{2}(x)$ from both sides to obtain the desired identity. Note that this subtlety will most likely be confusing to students, so whether you stress this point is up to you and where your class is.
Emphasize to students that there is no formula or algorithm for proving identities. It often takes experimentation using algebra and known identities.
Have students work through Problem 1 on the worksheet. This may be challenging for students, so make sure to go over the problem as a class.
Double-Angle Formulas
Introduce the idea of double-angle formulas: these will allow us to write the quantities $\cos(2\theta)$ and $\sin(2\theta)$ in terms of just $\cos(\theta)$ and $\sin(\theta)$. One option for introducing double-angle formulas is to derive $\sin(2\theta)$. If you choose to do this, a derivation is given below.
- We apply the law of sines to relate $2\theta$ and $\alpha$: $\frac{\sin(2\theta)}{2\sin(\theta)} = \frac{\sin(\alpha)}{1}$.
- We also know that $\sin(\alpha) = \cos(\theta)$ so we can conclude that $\frac{\sin(2\theta)}{2\sin(\theta)} = \cos(\theta)$, or
$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, the double-angle formula for sine.
Have students work through Problem 2 on the worksheet.
There is also a double-angle identity for cosine: $\cos(2\theta) = \cos^{2}(\theta)-\sin^{2}(\theta)$.
When proving identities, we often rely on previously-proven ones, such as the double-angle formulas. Encourage students to keep track of identities as they prove them, and to keep a master list somewhere in their notes for reference.
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