Using Trigonometric Identities: A Practical Guide
Learn to apply basic trigonometric identities to find exact values of trigonometric functions and simplify trigonometric expressions. This tutorial includes detailed examples with step-by-step solutions and practice exercises. For reference, a complete list of Fundamental Trigonometric Identities is available.
Examples with Detailed Solutions
Example 1: Finding Cosine from Sine
Given \( x \) is an angle in quadrant III and \( \sin x = -\frac{1}{3} \), find \( \cos x \).
Solution
- The Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) relates \( \sin x \) and \( \cos x \). Solving for \( \cos x \):
\[ \cos x = \pm \sqrt{1 - \sin^2 x} \]
- In quadrant III, \( \cos x \) is negative:
\[ \cos x = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \]
\[ = -\sqrt{1 - \frac{1}{9}} \]
\[ = -\sqrt{\frac{8}{9}} = -\frac{2\sqrt{2}}{3} \]
Verification: Check that \( \sin^2 x + \cos^2 x = 1 \) using the given \( \sin x \) and the calculated \( \cos x \).
Example 2: Finding Sine from Tangent
Given \( x \) is an angle in quadrant IV and \( \tan x = -5 \), find \( \sin x \).
Solution
- Using the identity \( 1 + \cot^2 x = \csc^2 x \) with reciprocal identities \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{1}{\tan x} \):
\[ 1 + \frac{1}{\tan^2 x} = \frac{1}{\sin^2 x} \]
- Solving for \( \sin x \):
\[ \sin x = \pm \sqrt{ \frac{\tan^2 x}{1 + \tan^2 x} } \]
- In quadrant IV, \( \sin x \) is negative. Substituting \( \tan x = -5 \):
\[ \sin x = -\sqrt{ \frac{(-5)^2}{1 + (-5)^2} } \]
\[ = -\sqrt{ \frac{25}{26} } = -\frac{5}{\sqrt{26}} \]
Example 3: Simplifying a Trigonometric Expression
Simplify the expression:
\[ (\sin x + \cos x)^2 + (\sin x - \cos x)^2 \]
Solution
- Expand each square:
\[ (\sin x + \cos x)^2 + (\sin x - \cos x)^2 = (\sin^2 x + \cos^2 x + 2\sin x \cos x) + (\sin^2 x + \cos^2 x - 2\sin x \cos x) \]
- Combine like terms:
\[ = 2\sin^2 x + 2\cos^2 x \]
- Factor out 2 and apply the Pythagorean identity:
\[ = 2(\sin^2 x + \cos^2 x) = 2(1) = 2 \]
Practice Exercises
- Given \( x \) is in quadrant II and \( \sin x = \frac{1}{5} \), find \( \cos x \) and \( \tan x \).
- Given \( x \) is in quadrant I and \( \cot x = 3 \), find \( \cos x \).
- Simplify the trigonometric expression: \( (\sin x + \cos x)(\sin x - \cos x) + 2\cos^2 x \).
Exercise Solutions
- \( \cos x = -\frac{2\sqrt{6}}{5} \), \( \tan x = -\frac{\sqrt{6}}{12} \)
- \( \cos x = \frac{3\sqrt{10}}{10} \)
- The expression simplifies to 1.
Additional Resources