Chapter 8: Trigonometric Equations and Applications. Section 8.1: Simple Trigonometric Equations

Solving Trigonometric Equations In Algebra, when we solve an equation, our goal is to isolate the variable on one side of the equation. To do this, we use operations that “undo” each other. In Trigonometry, when we solve equations, our goal is to isolate the angle, which is usually represented by θ or x. To do this, we use the inverse trig. functions.

Example 1: Find the values of θ between 0 and 2π for which sin θ = 0.5 (to the nearest degree). θ = sin θ = 30º *Use this as a reference angle in the quadrants where sine is positive.

*Sine is positive in the first and second quadrants. 30º 180º - 30º = 150º

So the two angles that have 0.5 as sine are 30º and 150º. Check these in your calculators: –Is sin 30º ≈ 0.5? –Is sin 150º ≈ 0.5?

Example 2: Solve cos θ = to the nearest degree. θ = cos *Since we are using the answer as a reference angle, do not put the negative into the calculator.* θ = 71.33º *In which quadrants are cosines negative?*

Solving Trigonometric Equations Cosine is negative in the second and third quadrants. 180º º = º 180º º = º

Solving Trigonometric Equations So the two angles that have as cosine are º and º. Check these in your calculators: –Is cos º ≈ -0.32? –Is cos º ≈ -0.32?

Example 3: Solve sin θ = 0.98 to the nearest degree. θ = sin θ = 78.52º *Use this as a reference angle in the quadrants where sine is positive.

*Sine is positive in the first and second quadrants º 180º º = º

So the two angles that have 0.98 as sine are 78.52º and º. Check these in your calculators: –Is sin 78.52º ≈ 0.98? –Is sin º ≈ 0.98?

Solving Trigonometric Equations *If the equation is not in terms of sin, cos, or tan, then you must substitute with those functions and then solve.*

Example 4: Solve in radians. a)sec x = 2.5 = 1.16 (quadrants?) = QI: 1.16 and QIV: 5.12 b)csc x = -1.4 =.796 (quadrants?) = QIII: 3.94 and QIV: 5.49

HOMEWORK (Day 1) pg. 299; 1 – 6 all, 13, 16, 17

Slope Recall: the four types of slopes. Draw an example in your notebooks of each of the four types… –positive slope –negative slope –no slope –undefined slope

Positive Slope

Negative Slope

No Slope

Undefined Slope

Slope Recall: The slope formulas… slope: m = point-slope equation: y – y 1 = m( x – x 1 ) y-intercept equation: y = mx + b

Inclination The inclination of a line is the angle, α where 0º ≤ α ≤ 180º, that is measured from the positive x -axis to the line. *We want the angle of inclination to be positive.*

Inclination Theorem For any line with inclination α and slope m, m = tan α, if α ≠ 90º If α = 90º, then the line is vertical and therefore does not have a defined slope.

Example 5: If a line has inclination angle of 120º and contains the point (2, 3), find the slope and the equation of the line. m = x + y = 6.46

Example 6: Find the inclination of the line 3x + 5y = 8. m = -3/5 Reference angle = 30.96° Inclination = °

Example 7: Find the inclination of the line joining the points (-3, -5) and (3, -3). m = 1/3 Inclination = 18.43°

HOMEWORK (Day 2) pg. 299 – 300; 22 – 30 even