1. Precalculus
Midterm Study Guide
You may use your Unit Circle and Chart.
1. Find the sine, cosine tangent, cosecant, secant, and cotangent of θ.
15
9
θ
3. The angle of elevation from the end of the shadow
to the top of the building is 70° and the distance is 180 feet.
a) Find the height of the building to the nearest foot.
b) Find the length of the shadow to the nearest foot.
15
4. Find csc (sin-1 /17).
21
5. Find tan (cos -1
/29).
6. Solve the triangle ABC if A = 120°, b = 10, and c = 5. (Find B, C, and a).
7. Solve the triangle ABC if A = 135°, b = 8, and c = 10. (Find B, C, and a).
8. State the Midline, Amplitude, Period, and Phase Shift of
y = sin (
x
/3 - π/2). Then graph the function.
x π
9. Find the asymptotes of y = csc ( /3 - /2). Then graph the function.
10. State the Midline, Amplitude, Period, and Phase Shift of
x
y = tan ( /2 - π) + 2. Find the asymptotes, and then graph the function.
11. State the Midline, Amplitude, Period, and Phase Shift of
x
y = 2 cos ( /2 - π) – 1. Then graph the function.
x
12. Find the asymptotes of y = 2 sec ( /2 - π) – 1. Then graph the function.
13. Find cot [sin-1 (-√3/2) + cos-1 (1/2)].
14. Simplify 1 – cos2 x
1 – sin2 x
15. If csc x cos x = 6, find the value of tan x.
16. If sec x sin x = 2, find the value of cot x.
17. Find tan (α - β) if sin α = 12/13 and cos β = 3/5.

2. Note: tan (α ± β) = tan α ± tan β .
1  tan α tan β
18. If csc θ = 5/3, find sin 2θ.
Note: sin 2θ = 2 sin θ cos θ.
19. Find sin π/8.
Note: sin α/2 = ±√(1 – cos α)
√2
20. Verify sin x cos x tan x + cos2 x = 1.
21. Verify (sin x – 1)(tan x + sec x) = – cos x.
22. Give the graphs of y = arcsin x, y = arccos x, and y = arctan x.
23. a) Graph the equation y = (x + 2)2 – 1.
b) What is the domain and range of the relation?
c) Is the relation a function?
d) Identify maxima, minima, or inflection points.
24. a) Graph the equation y = (x + 2)2.
b) What is the domain and range of the relation?
c) Is the relation a function?
d) Identify maxima, minima, or inflection points.
25. a) Find the inverse of f(x) = (x + 2)2 – 1.
b) Graph the inverse.
c) Is the inverse a function?
26. Write an equation of the line perpendicular to the graph of 5x – 3y + 6 = 0 that
passes through the point (3, 2).
27. Write an equation of the line parallel to the graph of 4y + 8 = x that passes
through the point (2, 3).

3. 28. Solve the system of equations algebraically.
3x – 4y = 10 y=¾x–5 2x + 3y = 19
-3x + 4y = 8 y = 4x – 18 7x – y = 9
29. a) If A = 4 7, find the determinant of A.
-1 3
b) Find the inverse of A, if it exists.
30. a) If A = 4 7 and B = 3 5, find A – B if possible.
-1 3 -2 1
b) Find BA if possible (Go Rebels!).
31. a) If A = 4 7, find AA.
-1 3
b) If B = 3 -1 2, can (AA)B be multiplied? If so, multiply.
-1 2 1
32. Give the equation for y = | x | translated 3 units down and reflected over the
x-axis.
33. Give the equation for the graph of y = x2 reflected over the x-axis and translated 2
units up.
34. Graph f(x) ≥ – x2 + 2.
35. Graph f(x) ≥ (x + 2)3 – 3.
36. The function f(x) = (x3 + 1) has a vertical asymptote at ___________________
(x2 – 1)
and a slant asymptote at _____________________________.
37. Write an equation of a function with a vertical asymptote at x = 2
and a hole at x = – 1.
38. f(x) = 1 has a vertical asymptote at ________________ and a horizontal
(x + 3)

4. asymptote at __________________.
39. a) Write an equation of a function with roots 5, 2, 2i, and – 2i.
b) How many times does the graph cross the x-axis?
40. a) Write a polynomial equation with roots 4, 1 + 2i, 1 – 2i.
b) How many times does the graph cross the x-axis?
41. Factor the quadratic 4x2 + 6x = 3.
42. Find all the factors of x3 + 2x2 – x – 2 = 0 if one of the factors is x + 1.
43. a) Find the remainder of (2x3 – x2 + x – 2) ÷ (x + 3).
b) Is (x + 3) a factor of (2x – x2 + x – 2)?
44. Find the value of b so that the remainder of (2x3 – x2 + x + b) ÷ (x – 1) is 0.
45. Find the value of k so that the remainder of (x3 + kx2 – x – 7) ÷ (x + 1) is 0.
46. Solve the radical equation.
9+√x–1=1
47. Solve the rational equation.
1 = x+3.
x 2x2
48. Factor the function into partial fractions
7x + 1 .
2
x + 2x – 3