Basic Math

Trigonometric identities | Eleventh Grade

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Trigonometric Identities

đź“Ś Introduction

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. These identities are fundamental tools for simplifying expressions, solving equations, and proving other mathematical relationships.

Complementary Angle Identities (Cofunction Identities)

Definition:

Two angles are complementary if their sum equals 90° or \( \frac{\pi}{2} \) radians. Cofunction identities relate trigonometric functions of complementary angles.

Key Concept: The "co" in cosine, cotangent, and cosecant stands for "complementary"!

Cofunction Identities in Degrees:

\( \sin(90° - \theta) = \cos \theta \)

\( \cos(90° - \theta) = \sin \theta \)

\( \tan(90° - \theta) = \cot \theta \)

\( \cot(90° - \theta) = \tan \theta \)

\( \sec(90° - \theta) = \csc \theta \)

\( \csc(90° - \theta) = \sec \theta \)

Cofunction Identities in Radians:

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

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

\( \tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta \)

đź“ť Examples - Cofunction Identities:

Example 1: Simplify \( \sin(30°) \) using cofunctions

\( \sin(30°) = \cos(90° - 30°) = \cos(60°) = \frac{1}{2} \)

Example 2: If \( \sin \theta = 0.6 \), find \( \cos(90° - \theta) \)

\( \cos(90° - \theta) = \sin \theta = 0.6 \)

Symmetry and Periodicity

Even and Odd Functions:

Even Functions (symmetric about y-axis):

\( \cos(-\theta) = \cos \theta \)

\( \sec(-\theta) = \sec \theta \)

Odd Functions (symmetric about origin):

\( \sin(-\theta) = -\sin \theta \)

\( \tan(-\theta) = -\tan \theta \)

\( \csc(-\theta) = -\csc \theta \)

\( \cot(-\theta) = -\cot \theta \)

Periodicity:

Trigonometric functions repeat their values at regular intervals:

Period = \( 2\pi \) (or 360°):

  • \( \sin(\theta + 2\pi) = \sin \theta \)
  • \( \cos(\theta + 2\pi) = \cos \theta \)
  • \( \sec(\theta + 2\pi) = \sec \theta \)
  • \( \csc(\theta + 2\pi) = \csc \theta \)

Period = \( \pi \) (or 180°):

  • \( \tan(\theta + \pi) = \tan \theta \)
  • \( \cot(\theta + \pi) = \cot \theta \)

Pythagorean Identities

The Three Pythagorean Identities:

1. Fundamental Identity:

\( \sin^2 \theta + \cos^2 \theta = 1 \)

Alternative forms:

  • \( \sin^2 \theta = 1 - \cos^2 \theta \)
  • \( \cos^2 \theta = 1 - \sin^2 \theta \)

2. Tangent-Secant Identity:

\( 1 + \tan^2 \theta = \sec^2 \theta \)

Alternative forms:

  • \( \tan^2 \theta = \sec^2 \theta - 1 \)
  • \( \sec^2 \theta - \tan^2 \theta = 1 \)

3. Cotangent-Cosecant Identity:

\( 1 + \cot^2 \theta = \csc^2 \theta \)

Alternative forms:

  • \( \cot^2 \theta = \csc^2 \theta - 1 \)
  • \( \csc^2 \theta - \cot^2 \theta = 1 \)

Reciprocal Identities

Six Reciprocal Relationships:

\( \sin \theta = \frac{1}{\csc \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \)

\( \cos \theta = \frac{1}{\sec \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta} \)

\( \tan \theta = \frac{1}{\cot \theta} \quad \text{and} \quad \cot \theta = \frac{1}{\tan \theta} \)

Quotient Identities

Tangent and Cotangent:

\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

đź“ť Example - Using Multiple Identities:

If \( \sin \theta = \frac{3}{5} \) and \( \theta \) is in Quadrant II, find \( \cos \theta \), \( \tan \theta \), and \( \sec \theta \).

Step 1: Find \( \cos \theta \) using Pythagorean identity:
\( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \)
\( \cos \theta = \pm\frac{4}{5} \)
In Quadrant II, cosine is negative: \( \cos \theta = -\frac{4}{5} \)

Step 2: Find \( \tan \theta \) using quotient identity:
\( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3/5}{-4/5} = -\frac{3}{4} \)

Step 3: Find \( \sec \theta \) using reciprocal identity:
\( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-4/5} = -\frac{5}{4} \)

Sum and Difference Identities

Sine Sum and Difference:

\( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

\( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

Cosine Sum and Difference:

\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

\( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

Tangent Sum and Difference:

\( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

\( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

đź“ť Example 1 - Finding Exact Values:

Find the exact value of \( \sin(75°) \) using sum identities.

Solution:
Write \( 75° = 45° + 30° \)
\( \sin(75°) = \sin(45° + 30°) \)
\( = \sin 45° \cos 30° + \cos 45° \sin 30° \)
\( = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \)
\( = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \)
\( = \frac{\sqrt{6} + \sqrt{2}}{4} \)

đź“ť Example 2 - Verifying Identity:

Verify: \( \cos(180° - \theta) = -\cos \theta \)

Using difference formula:
\( \cos(180° - \theta) = \cos 180° \cos \theta + \sin 180° \sin \theta \)
\( = (-1) \cos \theta + (0) \sin \theta \)
\( = -\cos \theta \) âś“

Solving Trigonometric Equations Using Identities

General Strategy:

  1. Use sum/difference identities to expand or simplify
  2. Apply Pythagorean identities to convert between functions
  3. Use reciprocal or quotient identities to simplify
  4. Factor if possible
  5. Solve the resulting equations
  6. Check solutions in the original equation

đź“ť Example - Solving with Sum Identity:

Solve: \( \sin(x + 45°) = \frac{\sqrt{2}}{2} \) for \( 0° \leq x < 360° \)

Method 1 - Direct approach:
\( x + 45° = 45° \) or \( x + 45° = 135° \)
\( x = 0° \) or \( x = 90° \)

Method 2 - Expand first:
\( \sin x \cos 45° + \cos x \sin 45° = \frac{\sqrt{2}}{2} \)
\( \frac{\sqrt{2}}{2}(\sin x + \cos x) = \frac{\sqrt{2}}{2} \)
\( \sin x + \cos x = 1 \)
Then solve for x

⚡ Quick Summary - Essential Identities

Identity TypeKey Formula
Pythagorean\( \sin^2\theta + \cos^2\theta = 1 \)
Cofunction\( \sin(90° - \theta) = \cos\theta \)
Reciprocal\( \csc\theta = \frac{1}{\sin\theta} \)
Quotient\( \tan\theta = \frac{\sin\theta}{\cos\theta} \)
Sum (Sine)\( \sin(A + B) = \sin A\cos B + \cos A\sin B \)
Sum (Cosine)\( \cos(A + B) = \cos A\cos B - \sin A\sin B \)

đź“š Memory Aids

Tips for Remembering Identities:

  • Cofunctions: "co" means complementary (adds to 90°)
  • Even functions: Cosine and Secant (contain 'c')
  • Odd functions: Sine, Tangent, Cosecant, Cotangent
  • Sum/Difference: Sine uses + when adding, Cosine uses - when adding
  • Pythagorean: Start with \( \sin^2 + \cos^2 = 1 \), divide by \( \cos^2 \) or \( \sin^2 \) for others

⚠️ Common Mistakes to Avoid

  • ❌ \( \sin(A + B) \neq \sin A + \sin B \) (must use the identity!)
  • ❌ \( \cos(A - B) \neq \cos A - \cos B \) (use proper formula)
  • ❌ Forgetting the sign in Pythagorean identities when taking square roots
  • ❌ Confusing \( \sin^2\theta \) with \( \sin(\theta^2) \)
  • ❌ Not checking which quadrant determines the sign
  • âś“ Always expand sum/difference expressions using identities
  • âś“ Remember: \( \sin^2\theta \) means \( (\sin\theta)^2 \)
  • âś“ Check quadrants when using Pythagorean identities
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