Trigonometry and 3-Space
Item Bank: Mathematics 122
These sample problems were prepared by a committee of NB high school teachers,
as part of a workshop held at the University of New Brunswick, March 13 -- 14, 2008.
One point problems.
1. Find the determinant of M.
 6 1
M 
 2 3 
2. Given the following matrix equation, what would be the corresponding system of
equations?
 3 4 5   a   17 
 2 1
0   b    12 

  

 4 2 1   c   10 
3x  4y  z  12
3. Find the x-intercept of the graph of the equation.
4. Explain, in one sentence, why this system of equations has infinitely many solutions.
2x
x
+
+
4y
2y
=
=
6
3
1
 1 2 3
 3
B

5. If matrix A  
and
matrix
 4  , what are the dimensions of the product AB?
 0 1 2 
6. Write the mapping rule for the given equation.
1
(y  3)  sin 3(x  30 )
2
7. Determine the period of the function given by
y  14 sin 12 (x  90 )  3
8. State the range of the function given by y=2sin x.
9. Determine the value of c in the equation using the given graph.
c(y - 2) = sin 
10. Find the angle  in the third quadrant with tan  3
11. If sin  = -5/13, in what quadrant(s) could  be located?
12. What is the smallest positive coterminal angle of 1200 ?
13. Convert 200 to radians.
14. Evaluate cos 2 4 
15. Evaluate sec 240.
16. Find cot if sin = a and cos = b.
17. Simplify: (sec  + 1)(sec  - 1).
18. What are the restrictions on the domain of y=tan x?
19. Sketch the plane given by 2x  3y  z  12 .
20. Express 130 as a radian measure.
21. Express
22. Simplify:
5
rads in degrees.
4
csc 
cot 
23. Determine the period of the function given by: y  13 sin 2(x  90 )  5 .
24. Given the graph of y = f(x), find the range of f.
25. Sketch the graph of y  csc  .
26. Find a solution to the equation cos   
3
with  in the second quadrant.
2
27. Determine the location of the asymptote closest to the y-axis in the graph of
y  tan 2(x  6 ) .
28. Convert the matrix equation to a system of equations.
 2 5  x  7 
 3 4   y    12 
29. Write the mapping rule for the sinusoidal function: 3(y  1)  cos(x  45 )
30. Find sin if cos  
3
and tan  0 .
2
4 0 
31. Find the determinant of L  
.
 4 4 
32. Determine the equation of the sinusoidal axis for the function defined by the graph
below.
33. Find matrix whose inverse does not exist.
 4 1
34. Find the inverse of M  
.
 5 2 
35. Simplify: cos 30  sin 60 .
36. Find the exact value: sin2 315  cos2 150 .
37. Determine an equation of the sinusoidal function given the graph -- teacher to supply
graph --.
Three point problems.
1. Solve the system of equations below using the inverse matrix method. Algebraic solution
required.
4x  7y  0
x
 2y  3
2. The total cost in dollars of a Nintendo Wii system is defined by the equation z=260+65x+40y,
where x represents the number of additional games and y represents the number of
additional attachments.
(a) Determine the intercepts of the cost equation. Sketch and label the graph of the cost
equation, using the intercepts.
(b) What is the equation of the trace line in the x-z plane?
3. Answer (a) and (b) with regard to the graph below.
(a) Find:
(i)
(ii)
(iii)
Amplitude
Vertical translation
Period
(b) Determine the equation in terms of the sine or cosine function.
4. Answer (a) and (b) with regard to the sinusoidal equation

1
y  4   cos 3(  90 )
2
(a) Give the mapping rule for the function.
(b) Graph the equation showing at least one complete cycle.
5. Solve sin 2  
3
4
with 0    360 .
6. Graph the equation in radians, labelling key points, asymptotes, local minima and maxima.
y  3  csc(x   )
1


7. Let f ( )   cos 3      4 .

2
2
(a) State the amplitude of f ( ) .
(b) State the horizontal translation of f ( ) .
(c) Rewrite f ( ) as a sine function.


8. Sketch the graph of y  5  tan 2  x   for 0  x  2 .

2
9. Prove the identity.
cot  1  sin 2 

tan  1  cos2 
 4 2 
 110
F

10. If K  
and
 3
 3 1
10
1
2

 , show that F is the inverse of K.
5
5
11. Find the exact value of sin105 .
12. Set up, but do not solve, asystem of equations that represents the following situation.
A cash box contains $1425 in $5 bills, $10 bills, and $20 bills. There is a total of 113 bills.
The number of $10 bills is less than the number of $20 bills.
13. Answer (a)--(c) with regard to the plane given by 2x  4y  z  20 .
(a) Find the z-intercept.
(b) Find the trace line when z=4.
(c) Sketch the plane.
14. Give a cosine equation and a sine equation for the function graphed below.
15. The of a sine function are given in the table below.
4
3
4
5
4
7
4
9
4
11
4
y 2
6
2
2
2
6
x

(a) Graph the function.
(b) State the amplitude, horizontal stretch, vertical translation, and phase shift.
(c) Write an equation for the sine function.
16. Solve 3cos(3 )  2  0 with 0    2 .
17. Prove the identity:
sin2   cot 2    cos2   tan2    
18. Solve 2 sin(2 )  1  0 with 0    360
19. Find the length of the shortest interval between two vertical asymptotes in the graph of:
1
y  2   csc(2x)
3
20. Using the general rotation matrix
 cos
 sin 
find the image coordinates of
 sin  
,
cos  
ABC ABC, A(0,0), B(2, 3),C(4,1) under a rotation of 60.
Five point problems.
1. A dietician is planning a meal using ingredients A, B and C.
a) Each gram of ingredient A contains 3 mg of fat, 4 mg of protein and 3 mg of carbohydrate.
b) Each gram of ingredient B contains 6 mg of fat, 1 mg of protein, and 7 mg of carbohydrate.
c) Each gram of ingredient C contains 2 mg of fat, 2 mg of protein, and 5 mg of carbohydrate.
If the meal must provide 36 mg of fat, 32 mg of protein, and 47 mg of carbohydrate, how
many grams of each ingredient should be used?
In your solution, let A represent the number of grams of ingredient A, B the number of
grams of ingredient B, and C the number of grams of ingredient C.
Algebraic solution required.
2. A ferris wheel has a diameter of 10 m. At t = 4 s, a rider is at the bottom, 1 m off the ground.
The rider is at the bottom again 18 seconds later.
a) Sketch a sinusoidal graph that represents the height of the rider at time t.
b) Write an equation to represent your graph in (a).
c) Use your answer to (b) to find the height of the rider at 60 s.
d) When is the rider at a height of 6 m for the third time?
3. Prove the identity:
(sin  cos )2  1  2sin cos
4. Solve:
(3cos  1)(cos  1)  0
with 0    360
5. Answer (a) through (c) given both A and B in quadrant 4, and sin A   45 ,cos A 
(d)
(e)
(f)
(g)
6.
Find cosA and sinB.
Evaluate sin(A-B).
Evaluate cos(2A).
Evaluate cot B.
Prove the identity:
tan  cot 2   tan  
2
sin 2
24
25
7.
Prove the identity:
sec2 x  6 tan x  7 tan x  4

sec 2 x  5
tan x  2
8.
Solve the system of equations. Algebraic solution required.
3x
 6y  2z  12
4x 
y
 2z  28
x  6y  3z  9
9. Suppose you have a water wheel with a radius of 5 m. The lower 2 m of teh wheel is
submerged, and there is a nail stuck in the wheel at the 3 o'clock position.
(a) Mark appropriate scales on the axes below. Then graph the height of the nail above
water through 2 rotations, assuming that the wheel rotates clockwise at a constant
rate.
(b) Let h  f ( ) be the function graphed in (a). Find a formula for h as a sine function.
(c) Use your answer to (b) to calculate the height of the nail at 150.
10.
(a) Solve sin 2  
3
, 0    2
4
(b) Simplify the rational expression.
x2  x  6
x2  4
 2
x3  x2
x  5x  4
11. A vending machine contains nickels, dimes, and quarters. The number of quarters is twice
the number of dimes. The coins have a total value of $25. If there are 140 coins in all, how
many of each type are there?
In your solution, let n represent the number of nickels, d the number of dimes, and q the
number of quarters.
12. Find the exact value of
13. Prove the identity:
2 cos180  sin 45
.
cos 30
1
1
2


.
1  sin x 1  sin x cos 2 x
14. Find all solutions to tan 2 x 
sin x
20
cos x
15. A ball is thrown in the air from a height of 1.5 m.
After 0.7 seconds, the ball's height is 3.2 m. At 2.1 s, it is 0.6 m.
(a) Find the quadratic function that describes the height of the ball at time t.
(b) Use your function to determine the height of the ball at 1.3 s.
16. The tower of a windmill (from the ground to the centre of the blades) is 20 m high. Each of
the blades is 10 m long. The blades make a full revolution every 2 seconds.
One day, a fly lands on the tip of a blade when it is at its highest point.
(a) How far does the fly travel during one full revolution?
(b) Mark appropriate scales and sketch a graph of the height of the fly at time t through two full
revolutions.
(c) Give two times (after landing on the blade) when the fly is exactly 15 m high.
17. Evaluate the expression, leaving your answer in simplified radical form.
sin( 3 )
sin( 34 )  2 cos( 76 )
18. Prove the identity:
sin(   sin(     sin2   sin2 
19. Solve: 2sin x cos x  cos x
20. Answer (a) and (b) with regard to the equation below.
y  5  tan 2(x  2 )
(a) Sketch the graph on 0  x  2 .
(b) Identify the period.
21. Triangle ABC with vertices A(0,0), B(2, 3),C(4,1) ), is reflected in the line y  x .
(a) Give the coordinates of the vertices of the image.
(b) Find a matrix that produces (by multiplication) the reflection in y  x .