🔄 Trig Identities & Equations

Master trigonometric identities and equation solving techniques. These are essential for AMC 12 and appear frequently in AMC 10.

🎯 Key Ideas

Identities: Equations true for all valid inputs. Use them to simplify expressions and solve equations.

Addition Formulas: Relate trig functions of sum/difference of angles to functions of individual angles.

Double/Half Angles: Special cases of addition formulas for $2\theta$ and $\frac{\theta}{2}$.

📊 Fundamental Identities

Pythagorean Identities

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

Reciprocal Identities

  • $\csc\theta = \frac{1}{\sin\theta}$
  • $\sec\theta = \frac{1}{\cos\theta}$
  • $\cot\theta = \frac{1}{\tan\theta}$

Quotient Identities

  • $\tan\theta = \frac{\sin\theta}{\cos\theta}$
  • $\cot\theta = \frac{\cos\theta}{\sin\theta}$

➕ Addition Formulas

Sine Addition

  • $\sin(A + B) = \sin A \cos B + \cos A \sin B$
  • $\sin(A - B) = \sin A \cos B - \cos A \sin B$

Cosine Addition

  • $\cos(A + B) = \cos A \cos B - \sin A \sin B$
  • $\cos(A - B) = \cos A \cos B + \sin A \sin B$

Tangent Addition

  • $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
  • $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

Example: Find exact value of $\sin 15°$

Solution:

  • $\sin 15° = \sin(45° - 30°) = \sin 45° \cos 30° - \cos 45° \sin 30°$
  • $= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$
  • $= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$

🔄 Double Angle Formulas

Sine Double Angle

  • $\sin(2\theta) = 2\sin\theta \cos\theta$

Cosine Double Angle

  • $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
  • $\cos(2\theta) = 2\cos^2\theta - 1$
  • $\cos(2\theta) = 1 - 2\sin^2\theta$

Tangent Double Angle

  • $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Example: If $\sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, find $\sin(2\theta)$

Solution:

  1. Find $\cos\theta$: Since $\sin^2\theta + \cos^2\theta = 1$:

    • $\cos^2\theta = 1 - \sin^2\theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}$
    • $\cos\theta = -\frac{4}{5}$ (negative in Quadrant II)

  2. Apply double angle formula:

    • $\sin(2\theta) = 2\sin\theta \cos\theta = 2 \cdot \frac{3}{5} \cdot \left(-\frac{4}{5}\right) = -\frac{24}{25}$

🔄 Half Angle Formulas

Sine Half Angle

  • $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$

Cosine Half Angle

  • $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$

Tangent Half Angle

  • $\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$

Pitfall: Sign Determination

The $\pm$ sign depends on the quadrant of $\frac{\theta}{2}$.

🔄 Sum-to-Product Formulas

Sine Sum-to-Product

  • $\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$
  • $\sin A - \sin B = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$

Cosine Sum-to-Product

  • $\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$
  • $\cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$

Example: Simplify $\sin 75° + \sin 15°$

Solution:

  • $\sin 75° + \sin 15° = 2\sin\left(\frac{75° + 15°}{2}\right)\cos\left(\frac{75° - 15°}{2}\right)$
  • $= 2\sin(45°)\cos(30°) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$

🔄 Inverse Trig Functions

Principal Values

  • $\arcsin x$: Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • $\arccos x$: Range $[0, \pi]$
  • $\arctan x$: Range $(-\frac{\pi}{2}, \frac{\pi}{2})$

Properties

  • $\sin(\arcsin x) = x$ (for $x \in [-1,1]$)
  • $\arcsin(\sin x) = x$ (for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$)
  • $\cos(\arccos x) = x$ (for $x \in [-1,1]$)
  • $\arccos(\cos x) = x$ (for $x \in [0, \pi]$)

Example: Find $\arcsin\left(\sin\frac{5\pi}{6}\right)$

Solution:

  • $\sin\frac{5\pi}{6} = \sin\frac{\pi}{6} = \frac{1}{2}$ (using reference angle)
  • But $\frac{5\pi}{6}$ is not in the range of $\arcsin$ $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • The angle in range with same sine is $\frac{\pi}{6}$
  • Answer: $\frac{\pi}{6}$

🎯 AMC-Style Worked Example

Problem: Solve $\sin(2x) = \cos(x)$ for $0 \leq x < 2\pi$.

Solution:

  1. Use double angle formula: $\sin(2x) = 2\sin x \cos x$
  2. Substitute: $2\sin x \cos x = \cos x$
  3. Rearrange: $2\sin x \cos x - \cos x = 0$
  4. Factor: $\cos x(2\sin x - 1) = 0$
  5. Solve each factor:
    • $\cos x = 0$ → $x = \frac{\pi}{2}, \frac{3\pi}{2}$
    • $2\sin x - 1 = 0$ → $\sin x = \frac{1}{2}$ → $x = \frac{\pi}{6}, \frac{5\pi}{6}$
  6. Check all solutions in range: All four values are in $[0, 2\pi)$

Answer: $x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{3\pi}{2}$

🔍 Common Traps & Fixes

Trap: Wrong signs in half-angle formulas

Fix: Always determine the quadrant of $\frac{\theta}{2}$ to get the correct sign.

Trap: Domain restrictions on inverse trig

Fix: Remember the principal value ranges and check if your angle is in range.

Trap: Forgetting to check all solutions

Fix: When solving trig equations, always check the specified interval for all solutions.

Trap: Confusing sum-to-product and product-to-sum

Fix: Sum-to-product: $\sin A + \sin B = \ldots$; Product-to-sum: $\sin A \sin B = \ldots$

📋 Quick Reference

Essential Addition Formulas

  • $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
  • $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
  • $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle Formulas

  • $\sin(2\theta) = 2\sin\theta \cos\theta$
  • $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
  • $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Sum-to-Product

  • $\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$
  • $\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$

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