Section 8-1 Simple Trigonometric Equations

Solving Trigonometric Equations The sine graph (on p. 295) illustrates that there are many solutions to the trigonometric equation sin x = 0.5. We know that are two particular solutions.

Solving Trigonometric Equations Since the period of sin x is 2π, we can add integral multiples of 2π to get the other solutions., where n is any integer, are the general solutions of sin x = 0.5.

Solving Trigonometric Equations When solving an equation such as sin x = 0.6, we can use a calculator or a table of values.

Solving Trig Equations To solve an equation involving a single trigonometric function: 1.Transform the equation so that the trig function is alone on one side of the equals sign. 2.Make a note of where the trig function is positive or negative. 3.Take the inverse of the trig function to find the reference angle.

Solving Trig Equations When we take the inverse of the trig function to find the reference angle. –Always take the inverse of the POSITIVE value. –This guarantees that the reference angle will be in quadrant I. sin θ =.6vs.sin θ = -.6

Solving Trig Equations Apply the reference angle formulas to find the answers for the equation. –Notice that the only formula that has changed is the one for Quadrant III. –Why??? Θ 180°- Θ π - Θ 180° + Θ π + Θ 360°- Θ 2π - Θ

Slope and Inclination The inclination of a line is the angle α, where 0° < α < 180°, that is measured from the positive x-axis to the line. That means that inclination is only in quadrants I and II.

Slope and Inclination The line below has inclination of 35°.

Slope and Inclination The line below has inclination of 155°. The theorem that follows states that the slope of a nonvertical line is the tangent of its inclination.

Theorem: For any line with slope m and inclination α, m = tan α if α ≠ 90°. If α = 90°, then the line has no slope. (The line is vertical.) The proof is on p

Example Solve for 0° < Θ < 360°. Give solutions to the nearest tenth of a degree.

Example Find the inclination of the line joining the points (-3, -5) and (3, -3).