University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2023
Views:

90

Pages:

12
Author:

Arielle Armstrong

**Trigonometric Function** Nov. 2nd 2021 [Image of a right-angled triangle with sides labeled a, b, and c, and angle θ] **Pythagoras Thm:** c² = a² + b² | Degrees | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° | |---|---|---|---|---|---|---|---|---| | Radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π | | sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 | 0 | -1 | 0 | | cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 | -1 | 0 | 1 | | tan θ | 0 | 1/√3 | 1 | √3 | - | 0 | - | 0 | | Function | Domain | Range | Even/Odd | 1-1 or not | |---|---|---|---|---| | sin θ | ℝ | [-1, 1] | odd | not | | cos θ | ℝ | [-1, 1] | even | not | | tan θ | ℝ - {π/2 + nπ} | ℝ | odd | not | **Example:** Find dom {cos(2x-1)} * 2 functions * f(x) = cos x * g(x) = 2x - 1 * fog(x) = cos(2x-1) Domain: ℝ ∩ ℝ = ℝ **Example:** Find dom {sin(√(x-5))} * 2 functions * f(x) = sin x * g(x) = √(x-5) * Domain: x-5 ≥ 0 * x ≥ 5 * Domain: [5, ∞) * ℝ ∩ [5, ∞) = [5, ∞) **Example:** Find the range of: 1. f(x) = 3 / (3 + cos x) * Range for cos x: [-1, 1] * -1 ≤ cos x ≤ 1 * +5 +5 +5 * 4 ≤ cos x + 5 ≤ 6 * 3/4 ≥ 1 / (cos x + 5) ≥ 1/6 **Range: [1/6, 3/4]** **Simplified:** * 3/4 ≥ 1 / (cos x + 5) ≥ 1/6 * 4 ≤ cos x + 5 ≤ 6 * -1 ≤ cos x ≤ 1 **A) sin(2cos^-1(x))** cos^-1(x) = y => cos(y) = x [Image of a right triangle with sides x, √(1-x^2), and hypotenuse 1] sin(2y) = 2 sin y cos y = 2x√(1-x^2) **B) sin(π + cos^-1(2x))** cos^-1(2x) = y => 2x = cos y [Image of a right triangle with sides 2x, √(1-4x^2), and hypotenuse 1] sin(π+y) = sinπ cos y + cosπ sin y = 0*cos y + (-1)sin y = -sin y = -√(1-4x^2)