Sum and Difference Formulas in Trigonometry

Sum and Difference Formulas

Introduction

The sum and difference formulas are essential tools in trigonometry, allowing us to find the exact values of trigonometric functions for non-standard angles by expressing them as sums or differences of common angles. These formulas are also used to simplify expressions and solve trigonometric equations.

Sum and Difference Formulas for Sine, Cosine, and Tangent

• Sine Formulas:

  • Sum: $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

  • Difference: $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$

  

• Cosine Formulas:

  • Sum: $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

  • Difference: $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$

  

• Tangent Formulas:

  • Sum: $\tan(\alpha + \beta) = \dfrac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

  • Difference: $\tan(\alpha - \beta) = \dfrac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

  

Applications of Sum and Difference Formulas

Finding Exact Values

These formulas are used to find the exact values of trigonometric functions for angles that are not standard, by expressing them as sums or differences of special angles (such as $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$).

• Example: Find $\sin 75^\circ$.

  • Express $75^\circ$ 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: $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$, $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$, $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$, $\sin 30^\circ = \dfrac{1}{2}$

  • So, $\sin 75^\circ = \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$

Simplifying Trigonometric Expressions

Sum and difference formulas can be used to simplify complex trigonometric expressions into simpler forms or to establish identities.

• Example: Simplify $\cos 35^\circ \cos 55^\circ - \sin 35^\circ \sin 55^\circ$.

  • Recognize the cosine difference formula: $\cos(A)\cos(B) - \sin(A)\sin(B) = \cos(A+B)$

  • So, $\cos 35^\circ \cos 55^\circ - \sin 35^\circ \sin 55^\circ = \cos(35^\circ + 55^\circ) = \cos 90^\circ = 0$

Establishing Trigonometric Identities

Using Sum and Difference Formulas

These formulas are also used to prove or establish trigonometric identities by rewriting one side of an equation in terms of the other.

• Example: Prove $\cos \alpha \cot \beta + \tan \alpha \cos \beta = \dfrac{\cos(\alpha + \beta)}{\sin \beta \cos \alpha}$

  • Rewrite $\cot \beta = \dfrac{\cos \beta}{\sin \beta}$ and $\tan \alpha = \dfrac{\sin \alpha}{\cos \alpha}$

  • Combine and simplify using sum and difference formulas as needed.

Solving Trigonometric Equations

Application in Equations

Sum and difference formulas can be used to solve trigonometric equations, especially when the equation involves sums or differences of angles.

• Example: Solve $\sin \theta + 3 \cos \theta = 1$

  • Isolate one trigonometric function and use identities or algebraic manipulation to solve for $\theta$.

Reference Table: Pythagorean Triples

Pythagorean triples are sets of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. They are useful in trigonometry for constructing right triangles with integer side lengths.

(3, 4, 5)	(5, 12, 13)	(7, 24, 25)
(8, 15, 17)	(9, 40, 41)	(11, 60, 61)
(12, 35, 37)	(13, 84, 85)	(15, 112, 113)
(16, 63, 65)	(17, 144, 145)	(19, 180, 181)
(20, 21, 29)	(20, 99, 101)	(21, 220, 221)

Additional info: The above table is included for reference, as Pythagorean triples are often used in trigonometric problems involving right triangles.