Trigonometric Equations Solve Equations Involving a Single Trig

Trigonometric Equations Solve Equations Involving a Single Trig Function

Checking if a Number is a Solution

Finding All Solutions of A Trig Equation l Remember, trigonometric functions are periodic. Therefore, there an infinite number of solutions to the equation. To list all of the answers, we will have to determine a formula.

Finding All Solutions of A Trig Equation l l l Tan q = 1 tan-1(tan q) = tan-1 (1) q = p/4 To find all of the solutions, we need to remember that the period of the tangent function is p. Therefore, the formula for all of the solutions is

Finding All Solutions of A Trig Equation l l cos q = 0 cos-1 (cos q) = cos-1 0 q=0 The period for cos is 2 p. Therefore, the formula for all answers is 0 ± 2 pk (k is an integer).

Finding All Solutions of A Trig Equation

Solving a Linear Trig Equation l Solve

Solving a Trig Equation l Solve the equation on the interval 0 ≤ θ ≤ 2 p

Solving a Trig Equation l Solve the equation on the interval 0 ≤ θ ≤ 2 p

Solving a Trig Equation

Solving a Trig Equation l The number of answers to a trig equation on the interval 0 ≤ θ ≤ 2 p will be double the number in front of θ. In other words, if the angle is 2 θ the number of answers is 4. If the angle is 3 θ the number of answers is 6. If the angle is 4 θ the number of answers is 8, etc. unless the answer is a quadrantal angle.

Solving a Trig Equation l Keep adding 2 p to the answers until you have the needed angles.

Solving a Trig Equation l Solve the equation on the interval 0 ≤ θ ≤ 2 p

Solving a Trig Equation

Solving a Trig Equation l Solve the equation on the interval 0 ≤ θ ≤ 2 p

Solving a Trig Equation with a Calculator l l l sin θ = 0. 4 sin-1 (sin θ) = sin-1 0. 4 θ =. 411, p -. 411 = 2. 73 sec θ = -4 1/cos θ = -4 cos θ = -¼ cos-1 (cos θ) = cos-1 (-¼) θ = 1. 82 Need to find reference angle because this is a quadrant II answer.

Solving a Trig Equation with a Calculator l l To find reference angle given a Quad II angle p – answer (p – 1. 82 = 1. 32) Now add p to this answer (p + 1. 32) θ = 4. 46

Snell’s Law of Refraction l Light, sound and other waves travel at different speeds, depending on the media (air, water, wood and so on) through which they pass. Suppose that light travels from a point A in one medium, where its speed is v 1, to a point B in another medium, where its speed is v 2. Angle θ 1 is called the angle of incidence and the angle θ 2 is the angle of refraction.

Snell’s Law of Refraction l Snell’s Law states that

Snell’s Law of Refraction Some indices of refraction are given in the table on page 512

Snell’s Law of Refraction l The index of refraction of light in passing from a vacuum into water is 1. 33. If the angle of incidence is 40 o, determine the angle of refraction.

Snell’s Law of Refraction

Solving Trig Equations l l l Tutorial Sample Problems Video Explanations