Sum, Difference, and Double-Angle Trigonometric Identities

Sum and Difference Identities

Introduction

Sum and difference identities are essential tools in trigonometry that allow us to express the sine, cosine, and tangent of sums or differences of angles in terms of the sines, cosines, and tangents of the individual angles. These identities are foundational for simplifying expressions, solving equations, and proving other trigonometric identities.

Cosine of a Sum or Difference

• Cosine of a Sum: The formula for the cosine of the sum of two angles is: $\cos(A + B) = \cos A \cos B - \sin A \sin B$

• Cosine of a Difference: The formula for the cosine of the difference of two angles is: $\cos(A - B) = \cos A \cos B + \sin A \sin B$

• Sign Switch: Note that the sign in the formula switches: a plus in the argument becomes a minus in the formula, and vice versa.

• Example: To find $\cos 75^\circ$, write $75^\circ = 45^\circ + 30^\circ$ and use the sum formula: $\cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ$

Sine of a Sum or Difference

• Sine of a Sum: The formula for the sine of the sum of two angles is: $\sin(A + B) = \sin A \cos B + \cos A \sin B$

• Sine of a Difference: The formula for the sine of the difference of two angles is: $\sin(A - B) = \sin A \cos B - \cos A \sin B$

• Sign Consistency: The sign in the formula matches the sign in the argument.

• Example: To write an expression such as $\sin(75^\circ)$ as a sum, use $75^\circ = 45^\circ + 30^\circ$ and apply the sum formula.

Tangent of a Sum or Difference

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

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

• Example: To find $\tan 75^\circ$, write $75^\circ = 45^\circ + 30^\circ$ and use the sum formula: $\tan 75^\circ = \dfrac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}$

Double-Angle Identities

Introduction

Double-angle identities are special cases of the sum identities where the two angles are equal. They are useful for simplifying expressions and solving trigonometric equations involving double angles.

• Sine Double-Angle: $\sin(2\theta) = 2\sin \theta \cos \theta$

• Cosine Double-Angle: $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$ Alternate forms: $\cos(2\theta) = 2\cos^2 \theta - 1$ $\cos(2\theta) = 1 - 2\sin^2 \theta$

• Tangent Double-Angle: $\tan(2\theta) = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$

• Example: Prove the identity $\sin(2\theta) = 2\sin \theta \cos \theta$ by using the sum formula for sine.

Solving Trigonometric Equations Using Identities

Introduction

Trigonometric equations can often be solved more easily by applying sum, difference, or double-angle identities to rewrite the equation in a simpler form. Solutions are typically found within a specified interval, such as $[0, 2\pi]$.

• Example: Solve $\sin(2x) = 0$ in the interval $[0, 2\pi]$. Set $2x = n\pi$, so $x = 0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}, 2\pi$.

• Graphical Solution: The graph of the function can help visualize the number and location of solutions within the interval.

• Exact Solutions: After solving graphically or algebraically, list all solutions in the given interval.

Summary Table: Sum, Difference, and Double-Angle Identities

Identity	Formula
Cosine of a Sum	$\cos(A + B) = \cos A \cos B - \sin A \sin B$
Cosine of a Difference	$\cos(A - B) = \cos A \cos B + \sin A \sin B$
Sine of a Sum	$\sin(A + B) = \sin A \cos B + \cos A \sin B$
Sine of a Difference	$\sin(A - B) = \sin A \cos B - \cos A \sin B$
Tangent of a Sum	$\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$
Tangent of a Difference	$\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}$
Sine Double-Angle	$\sin(2\theta) = 2\sin \theta \cos \theta$
Cosine Double-Angle	$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$
Tangent Double-Angle	$\tan(2\theta) = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$