9.3 Double-Angle, Half-Angle, and Reduction Formulas
4 min read•june 18, 2024
Double-angle and are powerful tools for simplifying trigonometric expressions. They allow us to express trig functions of doubled or halved angles in terms of the original angle, making calculations easier and more efficient.
help simplify trig expressions with angles outside the first quadrant. By converting these angles to their first-quadrant equivalents, we can work with simpler expressions and still get accurate results.
Double-Angle Formulas
Double-angle formulas for exact values
Express trigonometric functions of double angles in terms of the original angle
sin(2θ)=2sin(θ)cos(θ) doubles the angle and multiplies sine and cosine of the original angle
cos(2θ)=cos2(θ)−sin2(θ) expresses cosine of double angle as difference of squared cosine and sine
cos(2θ)=2cos2(θ)−1 alternative form using double angle cosine formula and Pythagorean identity
cos(2θ)=1−2sin2(θ) alternative form using double angle cosine formula and Pythagorean identity
tan(2θ)=1−tan2(θ)2tan(θ) expresses tangent of double angle as ratio involving tangent of original angle
Calculate exact values by substituting known angle into appropriate formula and simplifying
Verify trigonometric identities by substituting for one side and simplifying to match the other side
Example: cos(2θ)=1−2sin2(θ) can be verified using cos(2θ)=cos2(θ)−sin2(θ) and Pythagorean identity
Half-Angle Formulas
Half-angle formulas for precise values
Express trigonometric functions of half angles in terms of the original angle
sin(2θ)=±21−cos(θ) expresses sine of half angle using cosine of original angle
cos(2θ)=±21+cos(θ) expresses cosine of half angle using cosine of original angle
tan(2θ)=±1+cos(θ)1−cos(θ) expresses tangent of half angle using cosine of original angle
tan(2θ)=sin(θ)1−cos(θ)=1+cos(θ)sin(θ) alternative forms using sine and cosine of original angle
Determine precise trigonometric values by substituting known angle into appropriate formula and simplifying
Example: cos(3π)=21+cos(π)=21+(−1)=0
Sign of result depends on quadrant of half-angle
Positive in first and fourth quadrants, negative in second and third quadrants
Reduction Formulas
Reduction formulas for simplification
Simplify trigonometric expressions involving angles outside first quadrant (0∘≤θ≤90∘)
Second quadrant (90∘<θ≤180∘):
sin(θ)=sin(180∘−θ) sine is positive, use reference angle (180∘−θ)
cos(θ)=−cos(180∘−θ) cosine is negative, use reference angle with negative sign
tan(θ)=−tan(180∘−θ) tangent is negative, use reference angle with negative sign
Third quadrant (180∘<θ≤270∘):
sin(θ)=−sin(θ−180∘) sine is negative, subtract 180∘ from angle and add negative sign
cos(θ)=−cos(θ−180∘) cosine is negative, subtract 180∘ from angle and add negative sign
tan(θ)=tan(θ−180∘) tangent is positive, subtract 180∘ from angle
Fourth quadrant (270∘<θ≤360∘):
sin(θ)=−sin(360∘−θ) sine is negative, use reference angle (360∘−θ) with negative sign
cos(θ)=cos(360∘−θ) cosine is positive, use reference angle
tan(θ)=−tan(360∘−θ) tangent is negative, use reference angle with negative sign
Simplify complex trigonometric expressions by identifying quadrant of angle and applying appropriate formula to reduce angle to first quadrant
Example: sin(210∘)=−sin(210∘−180∘)=−sin(30∘)=−21
Comparison of trigonometric formulas
Double-angle formulas express trigonometric functions of double angles in terms of original angle
Useful for calculating exact values and verifying identities
Half-angle formulas express trigonometric functions of half angles in terms of original angle
Helpful for determining precise trigonometric values and solving equations
Reduction formulas simplify trigonometric expressions involving angles outside first quadrant by reducing angle to first quadrant
Makes expressions easier to evaluate and manipulate
Fundamental Concepts in Trigonometry
Key trigonometric concepts
Unit circle: A circle with radius 1 centered at the origin, used to define trigonometric functions for any angle
Radian measure: An alternative way to measure angles, where one radian is the angle subtended by an arc length equal to the radius of the circle
Periodicity: The property of trigonometric functions that repeat their values at regular intervals
Coterminal angles: Angles that share the same terminal side and differ by multiples of 360° (or 2π radians)
Key Terms to Review (4)
Double-angle formulas: Double-angle formulas are trigonometric identities that express trigonometric functions of double angles (e.g., $2\theta$) in terms of the functions of the original angle ($\theta$). They are useful for simplifying expressions and solving equations involving trigonometric functions.
Half-angle formulas: Half-angle formulas are trigonometric identities that express the sine, cosine, and tangent of half an angle in terms of the square root. These formulas are useful for simplifying expressions involving trigonometric functions.
Pythagorean Theorem: The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.
Reduction formulas: Reduction formulas rewrite trigonometric functions of multiple angles in terms of functions of simpler angles. They simplify complex expressions and are particularly useful for integration and solving equations.