Trigonometric Identities Study Notes

Trigonometric Identities

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.

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).

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}

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}).

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
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.

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).

Simplification Example:
- Simplifying \frac{\text{sec}^2 \theta}{\text{sec}^2 \theta - 1} to obtain \text{csc}^2 \theta
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)

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

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).

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.

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 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.