Trigonometric Identities Study Notes

Trigonometric Identities

Introduction

• Usage of the Book:

  • Simply reading the booklet is insufficient for learning.

  • Emphasis is placed on the need for students to use pencil and paper to work through examples before consulting solutions.

  • Completing exercises independently is crucial for solidifying understanding.

Objectives

• By completing the workbook, students should:

  • Be familiar with trigonometric functions: sin, cos, tan, sec, csc, and cot and understand their relationships.

  • Understand the identity: $ext{sin}^2 \theta + \text{cos}^2 \theta = 1$.

  • Know expressions for sin, cos, and tan of sums and differences of angles.

  • Be able to simplify expressions and verify identities involving trigonometric functions.

  • Know how to differentiate all trigonometric functions.

  • Be familiar with expressions for $\text{sin}(2\theta)$, $\text{cos}(2\theta)$, $\text{tan}(2\theta)$ and apply them to simplify trigonometric functions.

  • Reduce expressions involving powers and products of trigonometric functions to simpler forms suitable for integration.

Pretest

• This section assumes understanding of radian measure and definitions/properties of standard trigonometric functions.

• Pretest questions include:

  1. Convert to Radians:

     • 60° → $\frac{\pi}{3}$

     • 135° → $\frac{3\pi}{4}$

     • 270° → $\frac{3\pi}{2}$

  2. Convert to Degrees:

     • $\frac{\pi}{4}$ → 45°

     • $-\frac{3\pi}{2}$ → −270°

     • $\frac{2\pi}{3}$ → 120°

  3. Evaluate Trig Functions:

     • $\text{sin}(\frac{\pi}{2}) = 1$

     • $\text{cos}(\frac{3\pi}{2}) = 0$

     • $\text{tan}(\frac{3\pi}{4}) = -1$

     • $\text{sin}(\frac{7\pi}{6}) = -\frac{1}{2}$

     • $\text{cos}(\frac{5\pi}{3}) = \frac{1}{2}$

     • $\text{tan}(2\pi) = 0$

  4. Graphing:

     • Sketch the graph of $y = \text{cos}(x)$.

Relations Between Trigonometric Functions

• Definitions via Unit Circle:

  • $\text{sin} \theta = y$

  • $\text{cos} \theta = x$

  • $\text{tan} \theta = \frac{y}{x}$

• Definitions of other functions:

  • $\text{sec} \theta = \frac{1}{\text{cos} \theta}$

  • $\text{csc} \theta = \frac{1}{\text{sin} \theta}$

  • $\text{cot} \theta = \frac{1}{\text{tan} \theta}$

• Functional Relationships (Identities):

  • Basic functions can be expressed in terms of sin and cos:

  • $\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}$

  • $\text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta}$

Proof of Identities

• Identities vs Equations:

  • An identity is true for all values of $\theta$ (e.g., $\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}$ is an identity).

  • An equation holds true only for specific values of $\theta$ (e.g., $\text{sin} \theta = \text{cos} \theta$ only when $\theta = \frac{\pi}{4}$).

Examples of Simplification

1. Example 1:

   • Simplifying $\text{cos} x \text{tan} x$:
     $\text{cos} x \text{tan} x = \text{cos} x \times \frac{\text{sin} x}{\text{cos} x} = \text{sin} x$

2. Example 2:

   • Proving that $\text{sin} \theta + \text{tan} \theta \text{csc} \theta + \text{cot} \theta = \text{sin} \theta \text{tan} \theta$:

   • Start with the left side:

   • $\text{sin} \theta + \frac{\text{sin} \theta}{\text{cos} \theta} \cdot \frac{1}{\text{sin} \theta} + \frac{\text{cos} \theta}{\text{sin} \theta}$

   • Reduce and simplify to reach the right side.

The Pythagorean Identities

• Pythagorean Theorem Concept:

  • In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

  • For $\triangle OAB$ (where $x = \text{cos} \theta$ and $y = \text{sin} \theta$):
    $\text{cos}^2 \theta + \text{sin}^2 \theta = 1$ (Identity 6).

• Derived Identities:

  • Dividing by $\text{cos}^2 \theta$:
    $1 + \text{tan}^2 \theta = \text{sec}^2 \theta$ (Identity 7).

  • Dividing by $\text{sin}^2 \theta$:
    $\text{cot}^2 \theta + 1 = \text{csc}^2 \theta$ (Identity 8).

Examples of Pythagorean Identities

1. Simplification Example:

   • Simplifying $\frac{\text{sec}^2 \theta}{\text{sec}^2 \theta - 1}$ to obtain $\text{csc}^2 \theta$

2. Identity Verification Example:

   • Show that $1 - 2\text{cos}^2 \theta = \text{tan} \theta - \text{cot} \theta$:
     $\text{tan} \theta - \text{cot} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} - \frac{\text{cos} \theta}{\text{sin} \theta}$… (Proceed with simplifications)

Sums and Differences of Angles

• Key identities for angle addition and subtraction :

  1. Sine:

  • $\text{sin}(\alpha + \beta) = \text{sin} \alpha \text{cos} \beta + \text{cos} \alpha \text{sin} \beta$ (Identity 9).

  • $\text{sin}(\alpha - \beta) = \text{sin} \alpha \text{cos} \beta - \text{cos} \alpha \text{sin} \beta$ (Identity 11).

  1. Cosine:

  • $\text{cos}(\alpha + \beta) = \text{cos} \alpha \text{cos} \beta - \text{sin} \alpha \text{sin} \beta$ (Identity 10).

  • $\text{cos}(\alpha - \beta) = \text{cos} \alpha \text{cos} \beta + \text{sin} \alpha \text{sin} \beta$ (Identity 12).

Derivation of Sine and Cosine Identities

• Derivations based on congruent triangles and distances between points on the unit circle:

  • For $\text{cos}(\alpha - \beta)$, show distance relationships equal:

  $2 - 2\text{cos}(\alpha - \beta) = 2 - 2\text{cos} \alpha \text{cos} \beta - 2\text{sin} \alpha \text{sin} \beta.$

• Use coordinate geometry and congruency of triangles to find relationships between sine and cosine formulas.

Example Problems

1. Example: Simplifying sums and differences using derived identities, such as simplifying $\text{sin}(a + b) + \text{sin}(a - b)$.

Summary: Tangent Identities

• Tangent angle addition and subtraction formulas:

  • $\text{tan}(A + B) = \frac{\text{tan} A + \text{tan} B}{1 - \text{tan} A \text{tan} B}$ (Identity 13).

  • $\text{tan}(A - B) = \frac{\text{tan} A - \text{tan} B}{1 + \text{tan} A \text{tan} B}$ (Identity 14).

Double Angle Formulas

• Double angle formulas summarizing all primary functions:

  • $\text{sin}(2\theta) = 2 \text{sin} \theta \text{cos} \theta$ (Identity 15).

  • $\text{cos}(2\theta) = \text{cos}^2 \theta - \text{sin}^2 \theta$ (Identity 16).

  • Alternates:

  • $\text{cos}(2\theta) = 2\text{cos}^2 \theta - 1$ (Identity 17).

  • $\text{cos}(2\theta) = 1 - 2\text{sin}^2 \theta$ (Identity 18).

  • $\text{tan}(2\theta) = \frac{2 \text{tan} \theta}{1 - \text{tan}^2 \theta}$ (Identity 19).

Applications of Double Angle Formulas

• These identities help with integration, such as:

  • $\int \text{cos}^2 \theta d\theta = \int \frac{1}{2}(1 + \text{cos}(2\theta)) d\theta$.

Final Exercises

• Exercises include proofs and simplifications of the given trigonometric identities, such as:

  • Determine $\int \text{sin}(6x) \text{cos}(2x) dx$ using appropriate formulas.

Solutions to Exercises

• Solutions for pretests, exercises 1-8 are provided along with explanation summaries of each for quick reference.

• Examples aid in verifying correct understanding and application of the identities involved in calculus.