Double-Angle, Power-Reducing, and Half-Angle Formulas in Trigonometry 5.3

Double-Angle, Power-Reducing, and Half-Angle Formulas

Introduction

This section explores special trigonometric identities known as double-angle, power-reducing, and half-angle formulas. These formulas are essential tools for simplifying trigonometric expressions, solving equations, and evaluating exact values of trigonometric functions. Mastery of these identities is crucial for advanced mathematics, including calculus.

Double-Angle Formulas

Definition and Key Formulas

• Double-angle formulas express trigonometric functions of $2\theta$ in terms of functions of $\theta$.

• They are derived from the sum formulas for sine, cosine, and tangent.

Key Double-Angle Formulas:

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

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

• Alternate forms for cosine:

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

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

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

Example: Find $\sin(2\theta)$ if $\sin\theta = 0.6$ and $\cos\theta = -0.8$.

• $\sin(2\theta) = 2 \times 0.6 \times (-0.8) = -0.96$

Power-Reducing Formulas

Definition and Key Formulas

• Power-reducing formulas rewrite powers of sine, cosine, and tangent in terms of the first power of cosine of a double angle.

• These are especially useful in calculus for integrating powers of trigonometric functions.

Key Power-Reducing Formulas:

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

• $\cos^2\theta = \dfrac{1 + \cos(2\theta)}{2}$

• $\tan^2\theta = \dfrac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$

Example: Rewrite $\cos^4\theta$ in terms of first powers of cosine.

• First, $\cos^4\theta = (\cos^2\theta)^2$

• Apply the power-reducing formula twice:

• $\cos^2\theta = \dfrac{1 + \cos(2\theta)}{2}$, so $\cos^4\theta = \left(\dfrac{1 + \cos(2\theta)}{2}\right)^2$

• Expand and use the formula again for $\cos^2(2\theta)$:

• $\cos^4\theta = \dfrac{1}{4} + \dfrac{1}{2}\cos(2\theta) + \dfrac{1}{4}\cos^2(2\theta)$

• $\cos^2(2\theta) = \dfrac{1 + \cos(4\theta)}{2}$

• Final answer: $\cos^4\theta = \dfrac{3}{8} + \dfrac{1}{2}\cos(2\theta) + \dfrac{1}{8}\cos(4\theta)$

Half-Angle Formulas

Definition and Key Formulas

• Half-angle formulas express trigonometric functions of $\dfrac{\theta}{2}$ in terms of $\cos\theta$.

• The sign (plus or minus) depends on the quadrant in which $\dfrac{\theta}{2}$ lies.

Key Half-Angle Formulas:

• $\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}$

• $\cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}$

• $\tan\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{1 + \cos\theta}}$

• Alternative forms for tangent (no sign ambiguity):

  • $\tan\dfrac{\theta}{2} = \dfrac{\sin\theta}{1 + \cos\theta}$

  • $\tan\dfrac{\theta}{2} = \dfrac{1 - \cos\theta}{\sin\theta}$

Example: Compute $\sin(22.5^\circ)$ using a half-angle formula.

• $22.5^\circ = \dfrac{45^\circ}{2}$, so $\sin(22.5^\circ) = \sqrt{\dfrac{1 - \cos(45^\circ)}{2}} = \sqrt{\dfrac{1 - \dfrac{\sqrt{2}}{2}}{2}}$

• Simplify: $\sin(22.5^\circ) = \dfrac{\sqrt{2 - \sqrt{2}}}{2}$

Application Examples

Evaluating Trigonometric Functions Given a Value

Given $\cos\theta = -0.8$ and $90^\circ \leq \theta \leq 180^\circ$ (second quadrant):

• Draw a reference triangle: adjacent = $-0.8$, hypotenuse = $1$, opposite = $0.6$ (by Pythagoras).

• Find all six trigonometric functions:

  • $\sin\theta = 0.6$

  • $\cos\theta = -0.8$

  • $\tan\theta = -0.75$

  • $\csc\theta = 1.67$

  • $\sec\theta = -1.25$

  • $\cot\theta = -1.33$

• Use double-angle and half-angle formulas to find $\sin(2\theta)$, $\cos(2\theta)$, $\tan(2\theta)$, $\sin\dfrac{\theta}{2}$, $\cos\dfrac{\theta}{2}$, $\tan\dfrac{\theta}{2}$, etc.

Evaluating with Cotangent and Sine Information

Given $\cot\theta = 2$ and $\sin\theta < 0$ (third quadrant):

• Reference triangle: adjacent = $-1$, opposite = $-2$, hypotenuse = $\sqrt{5}$.

• Find all six trigonometric functions:

  • $\sin\theta = -\dfrac{2}{\sqrt{5}}$

  • $\cos\theta = -\dfrac{1}{\sqrt{5}}$

  • $\tan\theta = 2$

  • $\csc\theta = -\dfrac{\sqrt{5}}{2}$

  • $\sec\theta = -\sqrt{5}$

  • $\cot\theta = 0.5$

• Apply double-angle and half-angle formulas as above, paying attention to the quadrant for sign selection.

Using Double-Angle and Half-Angle Formulas for Exact Values

• To compute $\cos(165^\circ)$, write as $\cos\left(2 \times 82.5^\circ\right)$ or use the half-angle formula with $\theta = 330^\circ$:

• $\cos(165^\circ) = -\sqrt{\dfrac{1 + \cos(330^\circ)}{2}}$

• Since $\cos(330^\circ) = \dfrac{\sqrt{3}}{2}$, substitute and simplify.

Establishing Trigonometric Identities

Example: Proving an Identity

Establish the identity: $\dfrac{\tan\theta - \cot\theta}{\tan\theta + \cot\theta} = \cos(2\theta)$

• Rewrite in terms of sine and cosine:

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

• Substitute and simplify numerator and denominator to a single fraction.

• Result: $\dfrac{\sin^2\theta - \cos^2\theta}{\sin^2\theta + \cos^2\theta} = \cos(2\theta)$

Summary Table: Key Formulas

Formula Type	Sine	Cosine	Tangent
Double-Angle	$\sin(2\theta) = 2\sin\theta\cos\theta$	$\cos(2\theta) = \cos^2\theta - \sin^2\theta$ or $2\cos^2\theta - 1$ or $1 - 2\sin^2\theta$	$\tan(2\theta) = \dfrac{2\tan\theta}{1 - \tan^2\theta}$
Power-Reducing	$\sin^2\theta = \dfrac{1 - \cos(2\theta)}{2}$	$\cos^2\theta = \dfrac{1 + \cos(2\theta)}{2}$	$\tan^2\theta = \dfrac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$
Half-Angle	$\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}$	$\cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}$	$\tan\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{1 + \cos\theta}}$ or $\dfrac{\sin\theta}{1 + \cos\theta}$ or $\dfrac{1 - \cos\theta}{\sin\theta}$

Additional info:

• When using half-angle formulas, always determine the correct sign based on the quadrant of $\dfrac{\theta}{2}$.

• These identities are foundational for integration techniques in calculus and for simplifying trigonometric expressions in advanced mathematics.