5.5 Trigonometric identities and angle formulas

Identities are equations true for every input in their common domain. They allow expressions to be rewritten and exact values to be calculated.

5.5.1 Fundamental identities

Pythagorean Identities. Wherever defined, \[\sin^2\theta+\cos^2\theta=1,\quad \tan^2\theta+1=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta.\]

Quotient identities are \(\tan\theta=\sin\theta/\cos\theta\) and \(\cot\theta=\cos\theta/\sin\theta\). To verify an identity, transform one side into the other using valid algebra rather than manipulating both sides toward an unknown middle.

Worked Example: Rewriting tangent

If \(\theta\) is in quadrant III and \(\cos\theta=c<0\), then \(\sin\theta=-\sqrt{1-c^2}\) and \[\tan\theta=\frac{\sqrt{1-c^2}}{-c}.\]

Try It. Express \(\sin\theta\) in terms of positive \(\cos\theta=c\) in quadrant IV.
Check Your Work \(\sin\theta=-\sqrt{1-c^2}\).

Worked Example: Verifying an identity

\[\frac{\cos\theta}{1-\sin\theta}\cdot\frac{1+\sin\theta}{1+\sin\theta}=\frac{1+\sin\theta}{\cos\theta}=\sec\theta+\tan\theta.\]

Try It. Verify \(\cos\theta(\sec\theta-\cos\theta)=\sin^2\theta\).
Check Your Work The left side is \(1-\cos^2\theta=\sin^2\theta\).

5.5.2 Even, odd, addition, and subtraction formulas

Sine and tangent are odd, while cosine is even. The addition formulas include \[\sin(s+t)=\sin s\cos t+\cos s\sin t,\] \[\cos(s+t)=\cos s\cos t-\sin s\sin t.\] Replace \(t\) by \(-t\) for subtraction formulas.

Worked Example: Finding an exact compound angle

Using \(75^\circ=45^\circ+30^\circ\), \[\sin75^\circ=\frac{\sqrt2}{2}\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\frac12=\frac{\sqrt6+\sqrt2}{4}.\]

Try It. Find \(\cos75^\circ\) exactly.
Check Your Work \((\sqrt6-\sqrt2)/4\).

Worked Example: Using odd symmetry

\(\sin(-40^\circ)=-\sin40^\circ\), while \(\cos(-40^\circ)=\cos40^\circ\).

Try It. Simplify \(\tan(-x)\).
Check Your Work \(-\tan x\).
Applied Problem: Combining two directional changes. A survey direction is formed by adding \(45^\circ\) and \(30^\circ\). Use angle formulas to find its exact sine and cosine, then confirm that their squares sum to 1.
Check Your Work \(\sin75^\circ=(\sqrt6+\sqrt2)/4\) and \(\cos75^\circ=(\sqrt6-\sqrt2)/4\). Squaring and adding gives 1.

5.5.3 Practice Problems

  1. Simplify \(1-\sin^2x\).
    Check Your Work \(\cos^2x\).
  2. Rewrite \(\tan x\) using sine and cosine.
    Check Your Work \(\sin x/\cos x\).
  3. Verify \(\sec^2x-\tan^2x=1\).
    Check Your Work It is a rearrangement of \(1+\tan^2x=\sec^2x\).
  4. Find \(\cos15^\circ\) exactly.
    Check Your Work \((\sqrt6+\sqrt2)/4\).
  5. Simplify \(\cos(-x)\).
    Check Your Work \(\cos x\).
  6. If a pipe direction is \(-25^\circ\), relate its sine to \(\sin25^\circ\).
    Check Your Work \(\sin(-25^\circ)=-\sin25^\circ\).
  7. A resultant direction is \(105^\circ=60^\circ+45^\circ\). Find its exact cosine.
    Check Your Work \((\sqrt2-\sqrt6)/4\).
  8. If \(\sin\theta=3/5\) in quadrant II, use an identity to find tangent.
    Check Your Work \(\cos\theta=-4/5\), so \(\tan\theta=-3/4\).