Double-angle and half-angle formulas are powerful tools for simplifying trigonometric expressions. They allow you to express trigonometric functions of double or half angles in terms of the original angle, making complex calculations more manageable.

Reduction formulas complement these by simplifying expressions with large angles. Together, these techniques form a versatile toolkit for solving trigonometric problems and verifying identities, enhancing your ability to work with periodic functions.

Double-Angle Formulas

Apply double-angle formulas to calculate exact trigonometric values and verify identities

• Express trigonometric functions of double angles $\sin(2\theta)$, $\cos(2\theta)$, $\tan(2\theta)$ in terms of trigonometric functions of the original angle $\theta$
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
    • Alternatively express as $\cos(2\theta) = 2\cos^2(\theta) - 1$ or $\cos(2\theta) = 1 - 2\sin^2(\theta)$
  • $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$
• Calculate exact trigonometric values using double-angle formulas by identifying the original angle $\theta$ and the trigonometric function of the double angle, substituting known values of $\sin(\theta)$, $\cos(\theta)$, or $\tan(\theta)$ into the appropriate double-angle formula, and simplifying the resulting expression
• Verify identities using double-angle formulas by replacing trigonometric functions of double angles with their equivalent expressions, simplifying both sides of the identity independently, and comparing the simplified expressions to confirm equality

Half-Angle Formulas

Utilize half-angle formulas to determine precise trigonometric values in complex expressions

• Express trigonometric functions of half angles $\sin(\frac{\theta}{2})$, $\cos(\frac{\theta}{2})$, $\tan(\frac{\theta}{2})$ in terms of trigonometric functions of the original angle $\theta$
  • $\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$
  • $\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$
  • $\tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$ or alternatively express as $\tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}$
• Determine precise trigonometric values using half-angle formulas by identifying the original angle $\theta$ and the trigonometric function of the half angle, substituting the known value of $\cos(\theta)$ or $\sin(\theta)$ into the appropriate half-angle formula, simplifying the resulting expression, evaluating the square root, and considering the sign of the result based on the quadrant in which the half angle lies

Reduction Formulas

Implement reduction formulas to simplify trigonometric expressions and solve equations

• Simplify trigonometric expressions by reducing the angle to a smaller value within the first quadrant using common reduction formulas:
  • $\sin(\theta) = \cos(\frac{\pi}{2} - \theta)$
  • $\cos(\theta) = \sin(\frac{\pi}{2} - \theta)$
  • $\tan(\theta) = \cot(\frac{\pi}{2} - \theta)$
  • $\csc(\theta) = \sec(\frac{\pi}{2} - \theta)$
  • $\sec(\theta) = \csc(\frac{\pi}{2} - \theta)$
  • $\cot(\theta) = \tan(\frac{\pi}{2} - \theta)$
• Simplify trigonometric expressions using reduction formulas by identifying the trigonometric function and the angle in the expression, choosing the appropriate reduction formula, substituting the angle into the reduction formula, and simplifying the resulting expression using known trigonometric values or identities
• Solve equations using reduction formulas by applying reduction formulas to the trigonometric functions in the equation, simplifying the equation, isolating the variable, and solving for the variable using algebraic techniques
• Use radian measure when applying reduction formulas, as it simplifies calculations involving periodic functions

Applications and Comparisons

Compare and contrast the applications of angle formulas vs reduction formulas

• Double-angle formulas are useful for calculating exact values of trigonometric functions for double angles, verifying trigonometric identities involving double angles, and solving equations containing trigonometric functions of double angles
• Half-angle formulas are useful for determining precise values of trigonometric functions for half angles, simplifying complex expressions involving trigonometric functions of half angles, and solving equations containing trigonometric functions of half angles
• Reduction formulas are useful for simplifying trigonometric expressions by reducing the angle to a smaller value, solving equations involving trigonometric functions of angles outside the first quadrant, and expressing trigonometric functions in terms of other trigonometric functions
• Solve trigonometric problems efficiently by recognizing the appropriate formula to apply in a given situation, which may require a combination of double-angle, half-angle, and reduction formulas (example: simplifying an expression using a double-angle formula, then further simplifying the result with a reduction formula)

Fundamental Concepts

Understand the relationship between trigonometric functions and the unit circle

• The unit circle provides a visual representation of trigonometric functions and their relationships
• Periodic functions, such as sine and cosine, repeat their values at regular intervals, which is evident in the unit circle
• Even and odd functions can be identified using the unit circle, with cosine being an even function and sine being an odd function