Trigonometry LESSON FIVE - Trigonometric Equations Lesson Notes

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1 Example Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Primary Ratios Solving equations with the unit circle. a) b) c) 0 d) tan θ =

2 a) Example sinθ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. b) sinθ = - Primary Ratios Solving equations graphically with intersection points - - c) cosθ d) cosθ = e) tanθ f) tanθ = undefined

3 Example Find all angles in the domain 0 θ 60 that satisfy the given equation. Write the general solution. Primary Ratios Solving equations with a calculator. (degree mode) a) b) c)

4 Example 4 a) sinθ = Find all angles in the domain 0 θ that satisfy the given equation. Intersection Point(s) of Original Equation Primary Ratios Solving equations graphically with θ-intercepts. θ-intercepts b) cosθ = Intersection Point(s) of Original Equation θ-intercepts

5 Example 5 Solve a) non-graphically, using the cos - feature of a calculator. - 0 θ Primary Ratios Equations with b) non-graphically, using primary trig ratios the unit circle. c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

6 Example 6 a) non-graphically, using the sin - feature of a calculator. Solve sinθ = -0.0 θ ε R Primary Ratios Equations with primary trig ratios b) non-graphically, using the unit circle. c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

7 Example 7 Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Reciprocal Ratios Solving equations with the unit circle. a) b) c)

8 Example 8 a) θ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. b) θ Reciprocal Ratios Solving equations graphically with intersection points c) θ d) secθ = e) θ f) θ

9 Example 9 Find all angles in the domain 0 θ 60 that satisfy the given equation. Write the general solution Reciprocal Ratios Solving equations with a calculator. (degree mode) a) b) c)

10 Example 0 a) θ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Intersection Point(s) of Original Equation Reciprocal Ratios Solving equations graphically with θ-intercepts. θ-intercepts b) θ Intersection Point(s) of Original Equation θ-intercepts

11 Example a) non-graphically, using the sin - feature of a calculator. Solve cscθ = - 0 θ Reciprocal Ratios b) non-graphically, using the unit circle. Equations with reciprocal trig ratios c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

12 Example a) non-graphically, using the cos - feature of a calculator. Solve secθ = θ 60 b) non-graphically, using the unit circle. Reciprocal Ratios Equations with reciprocal trig ratios c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

13 Example Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. a) cosθ - = 0 b) θ First-Degree Trigonometric Equations c) tanθ - 5 = 0 d) 4secθ + = secθ +

14 Example 4 Find all angles in the domain 0 θ that satisfy the given equation. a) sinθcosθ = cosθ b) 7sinθ = 4sinθ First-Degree Trigonometric Equations Check the solution graphically. Check the solution graphically. - - c) sinθtanθ = sinθ d) tanθ + cosθtanθ = 0 Check the solution graphically. Check the solution graphically

15 Example 5 Find all angles in the domain 0 θ that satisfy the given equation. a) sin θ = b) 4cos θ - = 0 Second-Degree Trigonometric Equations Check the solution graphically. Check the solution graphically. - - c) cos θ = cosθ d) tan 4 θ - tan θ = 0 Check the solution graphically. Check the solution graphically

16 Example 6 a) sin θ - sinθ - = 0 Find all angles in the domain 0 θ that satisfy the given equation. Second-Degree Trigonometric Equations Check the solution graphically b) csc θ - cscθ + = 0 Check the solution graphically c) sin θ - 5sin θ + sinθ = 0 Check the solution graphically

17 Example 7 Solve each trigonometric equation. Double and Triple Angles a) θ 0 θ i) graphically: ii) non-graphically: - b) θ 0 θ i) graphically: ii) non-graphically: -

18 Example 8 Solve each trigonometric equation. Half and Quarter Angles a) θ 0 θ 4 i) graphically: ii) non-graphically: 4 - b) θ - 0 θ 8 i) graphically: ii) non-graphically:

19 Example 9 It takes the moon approximately 8 days to go through all of its phases. New Moon First Quarter Full Moon Last Quarter New Moon a) Write a function, P(t), that expresses the visible percentage of the moon as a function of time. Draw the graph. Visible % t b) In one cycle, for how many days is 60% or more of the moon s surface visible?

20 Example 0 Rotating Sprinkler N A rotating sprinkler is positioned 4 m away from the wall of a house. The wall is 8 m long. As the sprinkler rotates, the stream of water splashes the house d meters from point P. Note: North of point P is a positive distance, and south of point P is a negative distance. a) Write a tangent function, d(θ), that expresses the distance where the water splashes the wall as a function of the rotation angle θ. W S E θ P d b) Graph the function for one complete rotation of the sprinkler. Draw only the portion of the graph that actually corresponds to the wall being splashed. 8 4 d -4 θ -8 c) If the water splashes the wall.0 m north of point P, what is the angle of rotation (in degrees)?

21 Example Inverse Trigonometric Functions When we solve a trigonometric equation like cosx = -, one possible way to write the solution is: Inverse Trigonometric Functions Enrichment Example Students who plan on taking university calculus should complete this example. In this example, we will explore the inverse functions of sine and cosine to learn why taking an inverse actually yields the solution. a) When we draw the inverse of trigonometric graphs, it is helpful to use a grid that is labeled with both radians and integers. Briefly explain how this is helpful. y x

22 b) Draw the inverse function of each graph. State the domain and range of the original and inverse graphs (after restricting the domain of the original so the inverse is a function). y = sinx y 6 y = cosx y x x c) Is there more than one way to restrict the domain of the original graph so the inverse is a function? If there is, generalize the rule in a sentence. d) Using the inverse graphs from part (b), evaluate each of the following:

1.6 Applying Trig Functions to Angles of Rotation

1.6 Applying Trig Functions to Angles of Rotation wwwck1org Chapter 1 Right Triangles and an Introduction to Trigonometry 16 Applying Trig Functions to Angles of Rotation Learning Objectives Find the values of the six trigonometric functions for angles