# MCR3U Trigonometry and Sinusoidal Functions Test Chapters 5 and 6

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```					MCR3U Trigonometry and Sinusoidal Functions Test Chapters 5 and 6

1. [K 8]Consider the function                            .
a. Show an I/O diagram for this function.

b. State the period ________________, the amplitude __________________

the phase shift/horizontal translation ________________________ and the vertical shift/

vertical translation __________________________.
b. Graph the function for the interval         . Show how you found at least two points on the I/O
diagram.
y

x

2. [K 3 A 1]Two friends, Suban and Jas, start from the same point but ride their bikes down two different paths.
The paths diverge at an angle of 38°. Suban rides down his path at 12 km/h, and Jas rides down his path at 14
km/h. Determine how far apart the two are after riding at their respective speeds for 90 min.

3. [K 4]The point A(–3, –5) is on the terminal arm of an angle , in standard position.
a. Determine exact expressions for the secondary/reciprocal trigonometric ratios for the angle.
b. Determine possible values for angle     when                      .

4. [A 4]A rectangle has a diagonal of 8 cm. The diagonal creates a 60° angle at the base of the rectangle.
a) Write an exact expression for the base and the height of the rectangle.

b) Use your expressions to find the exact area of the rectangle.

5. [ A 3]Richard, Anton, and Donald are standing at three corners of a triangle in the middle of a park. Donald is
54 m from Anton, and Richard is 74 m from Donald. The angle at the vertex of the triangle where Richard is
standing is 34°. Calculate the possible distance(s) from Richard to Anton. How many solutions will there
be to this problem? Justify your answer. It is not necessary to solve the complete problem.

6. [K 4]Prove
7. [K 1 A 4] Sheryl is surveying a cliff to determine the elevation from the base of a canyon to the top of the
cliff. She lays out a line AB that is 225 m in length. She also sites a point C at the base of the cliff. Point D is
a point directly above point C, at the top of the cliff. She measures          to be 43°,         to be 58°, and
the angle of elevation from point A to point D to be 29°.
Draw a diagram to model this situation. Determine the height of the cliff.

8. [A 5]A Dutch windmill has blades that are 20 m in length, and the centre of their circular motion is a point 23
m above the ground. The blades have a frequency of 4 revolutions per minute when in operation.
a) Use a sinusoidal function to model the height above the ground of the tip of one blade as a function of
time.
b) How far off the ground is the tip of the blade at 10 s?
9. [T 3]Is there an angle that satisfies the equation                    ? If so, determine the value(s) for the
angle. If not, justify why not. Use the unit circle in your explanation. A solution dependent on a graphing
calculator will not earn marks.
MCR3U Trigonometry and Sinusoidal Functions Test Chapters 5 and 6

1. ANS:
a) The amplitude is 3.
b) The period is       , or 180
c) The phase shift is 30° to the right.
d) The vertical shift is 2 units down.
e)

PTS: 1                 DIF: 2                 REF: Knowledge and Understanding
OBJ: Section 5.3 | Section 5.4                LOC: D2.6 | D2.7 TOP: Trigonometric Functions
KEY: graph | amplitude | period | shifts | transformations | sine function
2. ANS:
After riding for 90 min (1.5 h), Suban has travelled          , or 18 km. Jas has travelled , or 21 km.
Let x represent the distance between the two friends.
Use the cosine law.

The riders are approximately 13.0 km apart.

PTS: 1                  DIF: 2                REF: Knowledge and Understanding | Application
OBJ: Section 4.4        LOC: D1.6             TOP: Trigonometric Functions
KEY: cosine law | two-dimensional problem
3. ANS:
The angle is in the third quadrant, therefore only the tangent ratio will be positive.
From the given point, x = –3 and y = –5.
Therefore,                                 and           .

PTS: 1             DIF: 2                REF: Knowledge and Understanding
OBJ: Section 4.2   LOC: D1.2             TOP: Trigonometric Functions
KEY: CAST rule | primary trigonometric ratios | reference angle | point on terminal arm
4. ANS:
a)                        and

b)

The area of the rectangle is      cm2.

PTS: 1               DIF: 3                REF: Knowledge and Understanding | Application
OBJ: Section 4.1     LOC: D1.1             TOP: Trigonometric Functions
KEY: special angles | primary trigonometric ratios | area
5. ANS:
a) Use A for Anton, D for Donald, and R for Richard.
Triangle 1                              Triangle 2

b) For triangle 1,                            For triangle 2,
In triangle 1, the distance from Richard to Anton is 96.0 m. In triangle 2, the distance from Richard to Anton
is 26.6 m.
c) There are two answers because two triangles can be drawn with the given information, each with a unique
solution. This is referred to as the ambiguous case.

PTS:   1            DIF: 3
REF:   Knowledge and Understanding | Application | Communication
OBJ:   Section 4.4  LOC: D1.6            TOP: Trigonometric Functions
KEY:   two-dimensional problem | ambiguous case
6. ANS:
a)

L.S. = R.S.

L.S. = R.S.

PTS: 1              DIF: 3                   REF: Knowledge and Understanding
OBJ: Section 4.6    LOC: D1.5                TOP: Trigonometric Functions
KEY: trigonometric identity
7. ANS:
a)
b) Calculate the length of BC first, using the sine law.
c) First, find the measure of C.

d) To find the height of the cliff, use the tangent ratio.
e)

The height of the cliff is approximately 86.6 m.

PTS: 1             DIF: 3                  REF: Knowledge and Understanding | Application
OBJ: Section 4.5   LOC: D1.7               TOP: Trigonometric Functions
KEY: primary trigonometric ratios | sine law | three-dimensional problem
8. ANS:
a)
b)
h

60

50

40

30

20

10

0.1   0.2   0.3   0.4   0.5   0.6   0.7   t

c) approximately 5.7 m

PTS: 1              DIF: 2                  REF: Knowledge and Understanding | Application
OBJ: Section 5.6    LOC: D3.2 | D3.3 TOP: Trigonometric Functions
KEY: model | sinusoidal function | graph | not angles
9. ANS:
a)

b)

c) This is verified using a calculator.
d)

PTS: 1               DIF: 2                  REF: Knowledge and Understanding
OBJ: Section 4.3     LOC: D1.4               TOP: Trigonometric Functions
KEY: reciprocal trigonometric ratios | unit circle

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