[9.1] Trigonometric Identities
...and the double-angle formula
Identities
The equation $$\frac{2}{1-x}=3$$ is true for only one value, $x=1/3$
But an identity, such as $$\sin^2\theta + \cos^2\theta = 1$$ is true for all possible angles $\theta$.
Examples
Use algebra to test whether this is an identity: $$\frac{\sin t}{1-\cos t}\stackrel{?}{=}\frac{1+\cos t}{\sin t}$$
Using graphs to check identities
Is this an identity or not? $$\cos\left(x+\frac \pi 3\right)\stackrel{?}{=} \cos x +\cos \frac \pi 3$$
Solution: graph each side. If it's an identity both sides should look the same:
$\stackrel{?}{=}$ |
Double-angle formula
$$\sin 2\theta \stackrel{?}{=}2\sin \theta$$
Nope!
But it is possible to find an expression for $\sin 2\theta$ with this triangle:
- Think about the big triangle,
- Apply the law of sines for the angle $\alpha$ and the angle $(\theta+\theta)$...
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.
.
$$\sin 2\theta=2 \sin \theta \cos \theta$$
Applying the law of cosines to the triangle, we get:
$$\cos 2\theta=1-2\sin^2\theta$$
Use Pythagoras to show that you could also write this as: $$\cos 2\theta = \cos^2\theta - \sin^2\theta.$$