Trigonometry Formulas: Sum, Difference, and Product Identities

The sum, difference, and product formulas for \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\) are essential tools in trigonometry. This guide provides detailed explanations, worked examples, and practice questions to help you master these identities.

Sum Formulas in Trigonometry

  1. \(\sin(x + y) = \sin x \cos y + \cos x \sin y\)
  2. \(\cos(x + y) = \cos x \cos y - \sin x \sin y\)
  3. \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\)

Example 1: Finding \(\sin(x + y)\)

Given \(\sin x = \frac{1}{5}\) and \(\sin y = -\frac{2}{3}\), with angle \(x\) in quadrant II and angle \(y\) in quadrant III, find the exact value of \(\sin(x + y)\).

Solution

Using the sum formula: \(\sin(x + y) = \sin x \cos y + \cos x \sin y\)

First, find \(\cos x\) using \(\sin^2 x + \cos^2 x = 1\):

\(\cos x = -\sqrt{1 - \sin^2 x} = -\sqrt{1 - \left(\frac{1}{5}\right)^2} = -\frac{\sqrt{24}}{5}\) (negative in quadrant II)

Next, find \(\cos y\):

\(\cos y = -\sqrt{1 - \sin^2 y} = -\sqrt{1 - \left(-\frac{2}{3}\right)^2} = -\frac{\sqrt{5}}{3}\) (negative in quadrant III)

Substitute values:

\(\sin(x + y) = \left(\frac{1}{5}\right)\left(-\frac{\sqrt{5}}{3}\right) + \left(-\frac{\sqrt{24}}{5}\right)\left(-\frac{2}{3}\right)\)

\(\sin(x + y) = \frac{-\sqrt{5} + \sqrt{24}}{15}\)

Difference Formulas in Trigonometry

  1. \(\sin(x - y) = \sin x \cos y - \cos x \sin y\)
  2. \(\cos(x - y) = \cos x \cos y + \sin x \sin y\)
  3. \(\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}\)

Example 2: Simplify \(\cos\left(x - \frac{\pi}{2}\right)\)

Solution

\(\cos\left(x - \frac{\pi}{2}\right) = \cos x \cos\frac{\pi}{2} + \sin x \sin\frac{\pi}{2}\)

Since \(\cos\frac{\pi}{2} = 0\) and \(\sin\frac{\pi}{2} = 1\):

\(\cos\left(x - \frac{\pi}{2}\right) = \sin x\)

Example 3: Find \(\sin(15^\circ)\) Exactly

Solution

\(15^\circ = 45^\circ - 30^\circ\), so:

\(\sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ\)

\(\sin(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)\)

\(\sin(15^\circ) = \frac{\sqrt{2}(\sqrt{3} - 1)}{4}\)

Product to Sum Formulas

  1. \(\sin x \cos y = \frac{1}{2}[\sin(x + y) + \sin(x - y)]\)
  2. \(\cos x \sin y = \frac{1}{2}[\sin(x + y) - \sin(x - y)]\)
  3. \(\cos x \cos y = \frac{1}{2}[\cos(x + y) + \cos(x - y)]\)
  4. \(\sin x \sin y = \frac{1}{2}[\cos(x - y) - \cos(x + y)]\)

Example 4: Simplify \(2\cos(3x)\cos(2x) - \cos(x)\)

Solution

Using formula 3: \(2\cos(3x)\cos(2x) = 2 \cdot \frac{1}{2}[\cos(5x) + \cos(x)]\)

So: \(2\cos(3x)\cos(2x) - \cos(x) = \cos(5x) + \cos(x) - \cos(x) = \cos(5x)\)

Sum to Product Formulas

  1. \(\sin x + \sin y = 2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right)\)
  2. \(\sin x - \sin y = 2\cos\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right)\)
  3. \(\cos x + \cos y = 2\cos\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right)\)
  4. \(\cos x - \cos y = -2\sin\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right)\)

Example 5: Factor \(\sin(4x) - \sin(2x)\)

Solution

Using formula 2: \(\sin(4x) - \sin(2x) = 2\cos\left(\frac{4x + 2x}{2}\right)\sin\left(\frac{4x - 2x}{2}\right)\)

\(\sin(4x) - \sin(2x) = 2\cos(3x)\sin(x)\)

Practice Questions

  1. Find the exact value of \(\sin(105^\circ)\)
  2. Factor the expression \(\cos(5x) - \cos(3x)\)
  3. Given \(\sin(x) = -\frac{1}{6}\) and \(\cos(y) = -\frac{1}{3}\) with both angles in quadrant III, find the exact value of \(\cos(x + y)\)
  4. Factor \(\cos(x) + \sin(x)\) [Hint: Convert to single trigonometric function first]

Solutions

  1. \(\sin(105^\circ) = \frac{\sqrt{2} + \sqrt{6}}{4}\)
  2. \(\cos(5x) - \cos(3x) = -2\sin(4x)\sin(x)\)
  3. \(\cos(x + y) = \frac{\sqrt{35} - \sqrt{8}}{18}\)
  4. \(\cos(x) + \sin(x) = \sqrt{2}\cos\left(x - \frac{\pi}{4}\right)\)

Additional Resources