Name Math 4 notes and problem set February 16, 2017 supplement to Chapter 4 page 1

Solving trigonometric equations Objective: Solve trigonometric equations using function graphs and circle diagrams.

Example 1: Solve sin(θ) = –0.5 Function graph approach: Graph y = sin(θ), graph y = –0.5, and find their intersections. There are many intersections, but in a single cycle there are two intersections, and then all the others can be found by moving left or right by multiples of 2π.

Circle diagram approach: An angle with a negative sine value must be in a quadrant where y is negative, so there are these two possible diagrams.

In the 3rd Quadrant diagram, you can make a reference triangle that turns out to be 30°-60°-90°, leading to θ = 210° = 7π/6 radians.

In the 4th Quadrant diagram, you can make a reference triangle that turns out to be 30°-60°-90°, leading to θ = 330° = 11π/6 radians.

There are also other angle measures co-terminal to these. For example, –30° = –π/6 radians is another way to reach the segment in the 4th Quadrant. Putting together a final answer: This answer could be obtained from either of the above methods but it’s best to pursue both methods and see them leading to the same answer.

In degrees: θ = 210° + 360° n or θ = 330° + 360° n where n stands for any integer

In radians: θ = 7π/6 + 2πn or θ = 11π/6 + 2πn where n stands for any integer

Note the use of 360° n or 2πn as a way of including all the co-terminal angles.

Name Math 4 notes and problem set February 16, 2017 supplement to Chapter 4 page 2

Example 2: Solve tan(θ) = –1 You fill in the missing details. Function graph approach:

Circle diagram approach: 1−=xy so x and y must have opposite signs:

In the 2nd Quadrant diagram: θ =

In the 4th Quadrant diagram: θ =

Put together a final answer:

Name Math 4 notes and problem set February 16, 2017 supplement to Chapter 4 page 3

Example 3: Solve sin(θ) = –0.62 Function graph approach: Graph y = sin(θ), graph y = –0.62, and find their intersections. This time the intersections don’t fall at simple multiples of π, but you can find use the calculator Intersect command to find them.

Alternate or additional method on calculator: sin–1(–0.62) = –0.67. Which intersection point is given by this value?

Circle diagram approach: An angle with a negative sine value must be in a quadrant where y is negative, so there are these two possible diagrams.

This time there are no special triangles to help, but you can use your knowledge of inverse trig functions and reference trianges to find the angles. Are they consistent with angles in the graph? Putting together a final answer:

θ = 3.81 + 2πn or θ = 5.61 + 2πn where n stands for any integer

Name Math 4 notes and problem set February 16, 2017 supplement to Chapter 4 page 4

Exercises The exercises have been kept to few in number so that you will have time to pursue each problem thoroughly using both the function graph approach and the circle diagram approach before stating your answers. Aim for a level of detail similar to what’s in the examples.

4. Solve the equation cos(θ) = 21 . (Follow a process similar to Example 1. Use radian

measures and give exact answers in terms of π.)

5. Solve the equation tan(θ) = 3 . (Use radian measures and give exact answers in terms of π.)

6. Solve the equation sin(θ) = 23

− . (Use degree measures and give exact answers.)

7. Solve the equation cos(θ) = –0.73. (Follow a process similar to Example 3. Give approximate answers using radian measures.)

8. Solve the equation tan(θ) = 5. (Follow a process similar to Example 3.)

9. Solve the equation sin(θ) = 1. (Something different happens in this problem. Give exact answers using radian measures.)

10. Solve the equation sec(θ) = –2. Hint: The secant function is reciprocal to the cosine function, so convert into a cosine equation, then solve that equation.