9.3 Double-Angle, Half-Angle, and Reduction Formulas

Double-angle formulas are powerful tools in trigonometry. They let you express trig functions of double angles using functions of the original angle. This simplifies complex expressions and helps solve tricky equations.

Reduction and half-angle formulas are equally useful. They allow you to rewrite trig functions of angles outside the first quadrant or express half-angles in terms of the original angle. These formulas are key for simplifying and solving trig problems.

Double-Angle Formulas

Double-angle formulas for exact values

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• Express trigonometric functions of double angles $(2\theta)$ using functions of the original angle $(\theta)$
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ product of sine and cosine of the original angle
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ difference of squared cosine and sine of the original angle
    • $\cos(2\theta) = 2\cos^2(\theta) - 1$ double the squared cosine minus 1
    • $\cos(2\theta) = 1 - 2\sin^2(\theta)$ 1 minus double the squared sine
  • $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$ ratio of double tangent to 1 minus squared tangent of the original angle
• Simplify expressions and solve trigonometric equations by applying double-angle formulas
  • Example: $\cos(2\theta) = \frac{1}{2}$ solve for $\theta$ using the appropriate double-angle formula

Reduction Formulas and Half-Angle Formulas

Reduction formulas in trigonometry

• Express trigonometric functions of angles outside the first quadrant using angles within the first quadrant
  • For any angle $\theta$:
    • $\sin(\theta \pm \pi) = -\sin(\theta)$ sine of an angle plus or minus $\pi$ equals the negative sine of the original angle
    • $\cos(\theta \pm \pi) = -\cos(\theta)$ cosine of an angle plus or minus $\pi$ equals the negative cosine of the original angle
    • $\tan(\theta \pm \pi) = \tan(\theta)$ tangent of an angle plus or minus $\pi$ equals the tangent of the original angle
  • For any angle $\theta$:
    • $\sin(\theta \pm \frac{\pi}{2}) = \pm\cos(\theta)$ sine of an angle plus or minus $\frac{\pi}{2}$ equals the positive or negative cosine of the original angle
    • $\cos(\theta \pm \frac{\pi}{2}) = \mp\sin(\theta)$ cosine of an angle plus or minus $\frac{\pi}{2}$ equals the negative or positive sine of the original angle
    • $\tan(\theta \pm \frac{\pi}{2}) = -\cot(\theta)$ tangent of an angle plus or minus $\frac{\pi}{2}$ equals the negative cotangent of the original angle
• Simplify trigonometric expressions and solve equations by applying reduction formulas
  • Example: $\sin(\frac{5\pi}{4}) = -\sin(\frac{\pi}{4})$ using the reduction formula $\sin(\theta + \pi) = -\sin(\theta)$

Half-angle formulas for precise values

• Express trigonometric functions of half angles $(\frac{\theta}{2})$ using functions of the original angle $(\theta)$
  • $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$ sine of half an angle equals the positive or negative square root of $(1 - \cos(\theta))$ divided by 2
  • $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$ cosine of half an angle equals the positive or negative square root of $(1 + \cos(\theta))$ divided by 2
  • $\tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}$ tangent of half an angle equals $(1 - \cos(\theta))$ divided by $\sin(\theta)$ or $\sin(\theta)$ divided by $(1 + \cos(\theta))$
• Determine the sign of half-angle expressions based on the quadrant of the original angle
  • Example: If $\theta$ is in Quadrant II, $\sin(\frac{\theta}{2})$ is positive, and $\cos(\frac{\theta}{2})$ is negative
• Calculate exact values and simplify trigonometric expressions using half-angle formulas
  • Example: Find the exact value of $\cos(\frac{\pi}{8})$ using the half-angle formula for cosine

Verification of trigonometric identities

• Verify trigonometric identities by combining double-angle, half-angle, and reduction formulas
  • Substitute appropriate formulas for trigonometric functions in the given identity
  • Simplify expressions on both sides of the equation using algebraic techniques
  • Verify that both sides of the equation are equal
• Recognize when to apply each type of formula based on the angles and trigonometric functions in the identity
  • Example: Verify the identity $\cos(2\theta) = 1 - 2\sin^2(\theta)$ by substituting the double-angle formula for cosine and simplifying

Fundamental Concepts in Trigonometry

The unit circle and angle measurement

• Understand the relationship between radians and degrees as units of angle measurement
• Use the unit circle to visualize trigonometric functions and their values
• Identify the signs of trigonometric functions in different quadrants of the coordinate plane

Periodic functions and trigonometric identities

• Recognize the periodic nature of trigonometric functions and their graphs
• Apply fundamental trigonometric identities to simplify expressions and solve equations