Math 112 Recitation Assignment Name: _________________________________ Week 8

Midterm #2 Review

Please complete these problems BEFORE your recitation week 8 recitation. Working through these problems (and discussing them with your classmates and TA) will be helpful in reviewing for the midterm. However, they should not be thought of as a guarantee of what will (or will not) be asked on the exam. Working through these problems should only be one component of your preparation for the exam. Reviewing lecture notes, recitation assignments and homework will also be important.

1. Evaluate 𝑐𝑜𝑠!! − !!

.

2. Solve the equation cos 2𝜃 = − !!

:

a. On the interval [0,2𝜋). b. On −∞,∞ . 3. Evaluate 𝑐𝑜𝑠!!  (𝑐𝑜𝑠 −0.75 ). Give results in both radians and degrees. Approximate to the

nearest hundredth. 4. Evaluate cos  (𝑐𝑜𝑠!! −0.75 ). Approximate to the nearest hundredth.

Midterm information:

When is the midterm?______________________________________________________________________

Where will YOU be taking the midterm? ______________________________________________________

What should you take with you to the midterm? _________________________________________________

Moon

Earth Sun

𝜃  

5. Find the phase angle 𝜃 for the phase angle 𝐹 = !!. Assume 0° ≤ 𝜃 ≤ 180° and use

𝐹 𝜃 = !!1 − 𝑐𝑜𝑠𝜃 . Then, Solve the equation for all 𝜃.

6. Solve the right triangle ABC if a = 15 and b = 8. 7. How many solutions does the equation 𝑡𝑎𝑛!! 3𝑥 = −1.37 have on the interval [0,2𝜋)? (A

graph may be helpful to determine this.) Solve the equation. Round your answer(s) to 3 decimal places.

8. Find the other trig functions, given:

a. 𝑐𝑠𝑐𝜃 = 2 and 𝑠𝑒𝑐𝜃 = − 2. b. 𝑐𝑜𝑠𝜃 = !!"

and 𝜃 in Quadrant IV.

A

B C a

c b

9. Simplify the expression:

1 − 𝑐𝑜𝑠! −𝜃sin  (−𝜃)𝑐𝑠𝑐𝜃

10. Verify each identity:

a.

€

cosθ1− sinθ

= secθ + tanθ b.

€

(cos x − sec x)2 = tan2 x − sin2 x

11. Write a trigonometric expression:

a. Write 𝑐𝑜𝑠𝜃 in terms of 𝑐𝑠𝑐𝜃 if 𝑡𝑎𝑛𝜃 < 0. b. Write 𝑐𝑜𝑠𝜃 in terms of x if 𝑐𝑠𝑐𝜃 = 𝑥 and 𝜃 is in Quadrant II.

12 a. Name two angles—one positive and one negative—that are contained in Quadrant IV and

have a reference angle of 37°.

b. Find the reference angle for 𝜃.

𝜃 = −85° 13. Solve the equation for 0 ≤ 𝜃 ≤ 2𝜋.

𝑐𝑜𝑠!𝜃 − 𝑐𝑜𝑠𝜃 = 0 14. Use a sum or difference identity to find the exact value of the expression (do not approximate

with a calculator.)

𝑠𝑖𝑛𝜋12

15. Find 𝑐𝑜𝑠 𝛼 + 𝛽 , given 𝑠𝑖𝑛𝛼 = !"!"

and 𝑐𝑜𝑠𝛽 = − !!"

, 𝛼 and 𝛽 are in Quadrant II.

Remember to study more stuff, too J