Problem 4 Solving a Cosine Equation

Suppose you want to find the time t in hours when the water level from Problem 3 is exactly 10 in. above the average level represented by f ( t ) = 0 . f open t close equals 0 .  What are all solutions to the equation − 30 cos ( 6 π 37 t ) = 10 negative 30 cosine . open . fraction 6 pi , over 37 end fraction , t . close . equals 10  in the interval from 0 to 25?

• Step 1 Use two equations. Graph y = 10 y equals 10  and y = − 30 cos ( 6 π 37 t ) y equals negative 30 cosine . open . fraction 6 pi , over 37 end fraction , t . close  on the same screen.

• Step 2 Use the INTERSECT feature to find the points at which the two graphs intersect.

  The graphs show four solutions in the interval. They are t ≈ 3.75 , 8.58 , 16.08 , and  20.92 . t almost equal to , 3.75 , comma , 8.58 , comma , 16.08 , comma , and , 20.92 , .

  The water level is 10 in. above the average level at about 3.75 h, 8.58 h, 16.08 h, and 20.92 h after 10:30 A.M.

  

✓ Got It?

4. What are all solutions to each equation in the interval from 0 to 2π?
   1. 3 cos 2 t = − 2 3 cosine 2 t equals negative 2
   2. − 2 cos θ = 1.2 negative 2 cosine theta equals 1.2
   3. Reasoning In the interval from 0 to 2π, when is − 2 cos θ negative 2 cosine theta  less than 1.2? Greater than 1.2?

✓ Lesson Check

Do you know HOW?

Sketch the graph of each function in the interval from 0 to 2π.

1. y = cos 1 2 θ y equals cosine , 1 half , theta
2. y = 2 cos π 3 θ y equals 2 cosine , pi over 3 , theta

Write a cosine function for each description. Assume that a > 0 . eh greater than 0 .

3. amplitude 3, period 2π
4. amplitude 1.5, period π

Do you UNDERSTAND?

5. Open-Ended Write a cosine function with amplitude 5 and between 2 and 3 cycles from 0 to 2π.
6. Assume θ is in the interval from 0 to 2π.
   1. For what values of θ is y positive for y = cos θ ? y equals cosine theta question mark
   2. For what values of θ is y positive for y = − sin θ ? y equals negative sine theta question mark
   3. Reasoning What sine function has the same graph as y = − 3 cos 2 π 3 θ ? y equals negative 3 cosine . fraction 2 pi , over 3 end fraction , theta . question mark