Sum and Difference Formulas in Trigonometry: Study Notes 5.2

Sum and Difference Formulas

Introduction

The sum and difference formulas are essential tools in trigonometry that allow us to find the exact values of trigonometric functions for angles that are not commonly found on the unit circle, such as 75°, 105°, or combinations of special angles. These formulas also play a crucial role in proving trigonometric identities and simplifying expressions.

Sum and Difference Formulas for Sine, Cosine, and Tangent

• Sine of a Sum: $\sin(x + y) = \sin x \cos y + \cos x \sin y$

• Sine of a Difference: $\sin(x - y) = \sin x \cos y - \cos x \sin y$

• Cosine of a Sum: $\cos(x + y) = \cos x \cos y - \sin x \sin y$

• Cosine of a Difference: $\cos(x - y) = \cos x \cos y + \sin x \sin y$

• Tangent of a Sum: $\tan(x + y) = \dfrac{\tan x + \tan y}{1 - \tan x \tan y}$

• Tangent of a Difference: $\tan(x - y) = \dfrac{\tan x - \tan y}{1 + \tan x \tan y}$

Applications of Sum and Difference Formulas

Finding Exact Values

These formulas allow us to compute the exact values of trigonometric functions for angles not directly available from the unit circle or special triangles.

• Example 1: Find $\sin 75^\circ$

  • Rewrite $75^\circ$ as $45^\circ + 30^\circ$.

  • Apply the sum formula: $\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

  • Substitute known values: $\dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$

• Example 2: Find $\cos 165^\circ$

  • Rewrite $165^\circ$ as $120^\circ + 45^\circ$.

  • Apply the sum formula: $\cos(120^\circ + 45^\circ) = \cos 120^\circ \cos 45^\circ - \sin 120^\circ \sin 45^\circ$

  • Substitute known values: $\left(-\dfrac{1}{2}\right) \cdot \dfrac{\sqrt{2}}{2} - \dfrac{\sqrt{3}}{2} \cdot \dfrac{\sqrt{2}}{2} = -\dfrac{\sqrt{2}}{4} - \dfrac{\sqrt{6}}{4} = -\dfrac{\sqrt{2} + \sqrt{6}}{4}$

• Example 3: Find $\tan 105^\circ$

  • Rewrite $105^\circ$ as $60^\circ + 45^\circ$.

  • Apply the sum formula: $\tan(60^\circ + 45^\circ) = \dfrac{\tan 60^\circ + \tan 45^\circ}{1 - \tan 60^\circ \tan 45^\circ}$

  • Substitute known values ($\tan 60^\circ = \sqrt{3}$, $\tan 45^\circ = 1$): $\dfrac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1}$

  • Rationalize and simplify: $\dfrac{\sqrt{3} + 1}{1 - \sqrt{3}} \cdot \dfrac{1 + \sqrt{3}}{1 + \sqrt{3}} = \dfrac{(\sqrt{3} + 1)^2}{(1 - \sqrt{3})(1 + \sqrt{3})} = \dfrac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}$

Combining Trigonometric Expressions

• Example 4: Write $\sin 20^\circ \cos 40^\circ + \cos 20^\circ \sin 40^\circ$ as a single trig value.

  • Recognize the sum formula for sine: $\sin x \cos y + \cos x \sin y = \sin(x + y)$

  • So, $\sin 20^\circ \cos 40^\circ + \cos 20^\circ \sin 40^\circ = \sin(20^\circ + 40^\circ) = \sin 60^\circ = \dfrac{\sqrt{3}}{2}$

• Example 5: Write $\cos\left(\dfrac{\pi}{5}\right) \cos\left(\dfrac{3\pi}{10}\right) - \sin\left(\dfrac{\pi}{5}\right) \sin\left(\dfrac{3\pi}{10}\right)$ as a single trig value.

  • Recognize the cosine sum formula: $\cos x \cos y - \sin x \sin y = \cos(x + y)$

  • So, $\cos\left(\dfrac{\pi}{5}\right) \cos\left(\dfrac{3\pi}{10}\right) - \sin\left(\dfrac{\pi}{5}\right) \sin\left(\dfrac{3\pi}{10}\right) = \cos\left(\dfrac{\pi}{5} + \dfrac{3\pi}{10}\right) = \cos\left(\dfrac{2\pi}{5}\right)$

• Example 6: Write $\dfrac{\tan 50^\circ - \tan 20^\circ}{1 + \tan 50^\circ \tan 20^\circ}$ as a single trig value.

  • Recognize the tangent difference formula: $\tan x - \tan y = \dfrac{\tan x - \tan y}{1 + \tan x \tan y}$

  • So, $\dfrac{\tan 50^\circ - \tan 20^\circ}{1 + \tan 50^\circ \tan 20^\circ} = \tan(50^\circ - 20^\circ) = \tan 30^\circ = \dfrac{1}{\sqrt{3}}$

Using Triangles to Find Missing Values

Solving for Unknown Sides

When given the sine or cosine of an angle and the quadrant, you can use the Pythagorean Theorem to find the other trigonometric values needed for the sum and difference formulas.

• Example 7: Given $\sin \alpha = \dfrac{12}{37}$ with $\alpha$ in Quadrant I, and $\sin \beta = \dfrac{12}{13}$ with $\beta$ in Quadrant II, find $\cos(\alpha + \beta)$.

  • Draw right triangles for each angle, using the given sine values to find the adjacent sides.

  • For $\alpha$ (Quadrant I): $\cos \alpha = \dfrac{35}{37}$ (since $\sqrt{37^2 - 12^2} = 35$)

  • For $\beta$ (Quadrant II): $\cos \beta = -\dfrac{5}{13}$ (since $\sqrt{13^2 - 12^2} = 5$, negative in Quadrant II)

  • Apply the formula: $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

  • Substitute values: $\dfrac{35}{37} \cdot \left(-\dfrac{5}{13}\right) - \dfrac{12}{37} \cdot \dfrac{12}{13} = -\dfrac{175}{481} - \dfrac{144}{481} = -\dfrac{319}{481}$

• Example 8: Given $\sin \alpha = \dfrac{3}{8}$ with $\alpha$ in Quadrant II, and $\cos \beta = \dfrac{12}{13}$ with $\beta$ in Quadrant IV, find $\sin(\alpha + \beta)$.

  • For $\alpha$ (Quadrant II): $\cos \alpha = -\dfrac{\sqrt{8^2 - 3^2}}{8} = -\dfrac{\sqrt{55}}{8}$

  • For $\beta$ (Quadrant IV): $\sin \beta = -\dfrac{5}{13}$ (since $\sqrt{13^2 - 12^2} = 5$, negative in Quadrant IV)

  • Apply the formula: $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

  • Substitute values: $\dfrac{3}{8} \cdot \dfrac{12}{13} + \left(-\dfrac{\sqrt{55}}{8}\right) \cdot \left(-\dfrac{5}{13}\right) = \dfrac{36}{104} + \dfrac{5\sqrt{55}}{104} = \dfrac{36 + 5\sqrt{55}}{104}$

Proving Trigonometric Identities Using Sum and Difference Formulas

Establishing Identities

Sum and difference formulas are powerful tools for proving trigonometric identities. The process often involves rewriting one side of the equation using these formulas and simplifying to show equality.

• Example 9: Prove $\sin\left(\dfrac{2\pi}{3} - \theta\right) = \dfrac{\sqrt{3}}{2} \cos \theta - \dfrac{1}{2} \sin \theta$

  • Apply the difference formula: $\sin\left(\dfrac{2\pi}{3} - \theta\right) = \sin\dfrac{2\pi}{3} \cos \theta - \cos\dfrac{2\pi}{3} \sin \theta$

  • Substitute values: $\sin\dfrac{2\pi}{3} = \dfrac{\sqrt{3}}{2}$, $\cos\dfrac{2\pi}{3} = -\dfrac{1}{2}$

  • So, $\dfrac{\sqrt{3}}{2} \cos \theta - (-\dfrac{1}{2}) \sin \theta = \dfrac{\sqrt{3}}{2} \cos \theta + \dfrac{1}{2} \sin \theta$

• Example 10: Prove $\tan(x + y) = \dfrac{\tan x + \tan y}{1 - \tan x \tan y}$

  • Start with $\tan(x + y) = \dfrac{\sin(x + y)}{\cos(x + y)}$

  • Apply sum formulas for sine and cosine:

  • $\tan(x + y) = \dfrac{\sin x \cos y + \cos x \sin y}{\cos x \cos y - \sin x \sin y}$

  • Divide numerator and denominator by $\cos x \cos y$:

  • $\tan(x + y) = \dfrac{\tan x + \tan y}{1 - \tan x \tan y}$

• Example 11: Prove $\cos(x + y) + \cos(x - y) = 2 \cos x \cos y$

  • Apply sum and difference formulas:

  • $\cos(x + y) = \cos x \cos y - \sin x \sin y$

  • $\cos(x - y) = \cos x \cos y + \sin x \sin y$

  • Add both sides: $[\cos x \cos y - \sin x \sin y] + [\cos x \cos y + \sin x \sin y] = 2 \cos x \cos y$

Summary Table: Sum and Difference Formulas

Function	Sum Formula	Difference Formula
Sine	$\sin(x + y) = \sin x \cos y + \cos x \sin y$	$\sin(x - y) = \sin x \cos y - \cos x \sin y$
Cosine	$\cos(x + y) = \cos x \cos y - \sin x \sin y$	$\cos(x - y) = \cos x \cos y + \sin x \sin y$
Tangent	$\tan(x + y) = \dfrac{\tan x + \tan y}{1 - \tan x \tan y}$	$\tan(x - y) = \dfrac{\tan x - \tan y}{1 + \tan x \tan y}$

Key Points

• Sum and difference formulas allow for the calculation of trigonometric values for non-standard angles.

• They are essential for simplifying expressions and proving identities in trigonometry.

• When given only one trigonometric value and a quadrant, use the Pythagorean Theorem to find the other values needed.

• Always pay attention to the sign of the trigonometric function based on the quadrant.

Additional info: The examples above demonstrate the use of sum and difference formulas for sine, cosine, and tangent, including how to combine expressions, find exact values, and prove identities. These skills are foundational for further study in trigonometry and calculus.