Topic 3.12 Notes – Equivalent Representations of Trigonometric Functions

What Equivalent Trigonometric Forms Are

Two trig expressions are equivalent when they match on their common domain. That domain part matters. Some rewrites introduce restrictions.

The whole point of rewriting is to make something easier to see, like:

• a constant, such as $\sin^2\theta+\cos^2\theta=1$
• one trig function instead of two
• a factorable or quadratic form
• exact values at awkward angles
• amplitude or range

The main source identities are

$$\sin^2\theta+\cos^2\theta=1$$

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$

$$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$$

A quick notation reminder. $\sin^2\theta$ means $(\sin\theta)^2$, not $\sin(\theta^2)$.

On the unit circle, a point $(x,y)$ gives $\cos\theta=x$ and $\sin\theta=y$, so the right triangle in the diagram leads directly to the Pythagorean identity.

Unit circle triangle and Pythagorean identity

The difference and double-angle identities come from the sum identities, so this is one connected system, not a random formula list.

Identities You Should Be Ready to Generate and Use

From the Pythagorean identity, you should be able to swap mixed squares for $1$, or isolate one square:

• $\sin^2\theta=1-\cos^2\theta$
• $\cos^2\theta=1-\sin^2\theta$
• $\tan^2\theta+1=\sec^2\theta$ when $\cos\theta\ne0$
• $1+\cot^2\theta=\csc^2\theta$ when $\sin\theta\ne0$

If $\sin\theta=x$, then

$$\cos\theta=\pm\sqrt{1-x^2}$$

The sign depends on the quadrant or given interval.

From the sum identities, you get:

• Difference identities
  • $\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$
  • $\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$
  • $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$, $\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$
• Double-angle identities
  • $\sin(2\alpha)=2\sin\alpha\cos\alpha$
  • $\cos(2\alpha)=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1$
• Power-reduction identities
  • $\sin^2\alpha=\dfrac{1-\cos(2\alpha)}{2}$
  • $\cos^2\alpha=\dfrac{1+\cos(2\alpha)}{2}$

Exact-value example. Since $75^\circ=45^\circ+30^\circ$,

$$\sin 75^\circ=\frac{\sqrt6+\sqrt2}{4}$$

and the same idea works for $5\pi/12=\pi/4+\pi/6$.

A derived identity you may see is the tangent sum identity:

$$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$

Also know how to combine $A\sin\theta+B\cos\theta$ into one sinusoid:

$$A\sin\theta+B\cos\theta=R\sin(\theta+\phi), \quad R=\sqrt{A^2+B^2}$$

This is great for reading amplitude and range quickly.

Choosing the Best Rewrite

Ask what form makes the goal easiest.

• If you see $\sin^2\theta+\cos^2\theta$, turn it into $1$.
• If everything else is in sine, use $\cos(2\theta)=1-2\sin^2\theta$.
• If everything else is in cosine, use $\cos(2\theta)=2\cos^2\theta-1$.
• If the angle is weird like $75^\circ$, use sum or difference identities.
• If you have $A\sin\theta+B\cos\theta$, combine into one sinusoid with $R=\sqrt{A^2+B^2}$.

A good rewrite can reveal zeros, extrema, range, or a solvable quadratic.

Verifying Identities and Solving Equations

Verifying identities

Work from one side only, usually the messier side, until it becomes the other side.

Useful moves:

1. rewrite everything in sine and cosine
2. use Pythagorean or double-angle identities
3. factor or get common denominators
4. keep domain restrictions in mind

Solving equations and inequalities

Rewrite first, then solve the easier form.

For $\cos(2\theta)=\sin\theta$ on $0\le\theta<2\pi$:

$$1-2\sin^2\theta=\sin\theta$$

$$2\sin^2\theta+\sin\theta-1=0$$

$$(2\sin\theta-1)(\sin\theta+1)=0$$

So $\sin\theta=\frac12$ or $\sin\theta=-1$, giving

$$\theta=\frac{\pi}{6},\ \frac{5\pi}{6},\ \frac{3\pi}{2}$$

For $\cos(2\theta)\le0$ on $0\le\theta<2\pi$, use the unit circle or cosine graph to get

$$\theta\in\left[\frac{\pi}{4},\frac{3\pi}{4}\right]\cup\left[\frac{5\pi}{4},\frac{7\pi}{4}\right]$$

Common Mistakes and Domain Traps

• $\sin(\alpha+\beta)\ne\sin\alpha+\sin\beta$ and $\cos(\alpha+\beta)\ne\cos\alpha+\cos\beta$
• Square roots need $\pm$, unless the interval fixes the sign.
• $\tan,\sec$ need $\cos\theta\ne0$. $\cot,\csc$ need $\sin\theta\ne0$.
• Canceling a factor that could be zero can lose solutions.
• Squaring or multiplying can create extraneous solutions, so check in the original equation.
• Two expressions may agree only on a restricted common domain.
• Exact-value work falls apart if you mix degrees and radians.