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7.2 – The Sine Ratio
A. Sine Ratio Introduction:
Another relationship exists between the ratio of the length of the side opposite a given
angle to the length of the hypotenuse. In two similar triangles this ratio will always be
the same. It is known as the __________________________.
sin  
length of side opposite 
length of hypotenuse
sin  
Example 1: Use your scientific calculator to determine the following sine ratios.
Round to four decimal places.
Note: make sure your calculator is in degree mode.
a) sin 10º
c) sin 50º
b) sin 45º
d) sin 70º
Example 2: Calculate the angle to the nearest degree.
a) sin A = 0.5164
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b) sin B = 0.8461
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B. Labelling Triangles
In order to solve angles and side in right triangles, it is important to be able to label them
properly. The three sides are: ______________________, ______________________,
______________________,
The _________________ is always directly opposite the right angle and it is the longest
side.
The ______________ side is always next to the angle of interest.
The _______________ side is always directly across the angle of interest.
A
Example 3: Consider ∆ABC
θ
Label all of the sides given θ and ϕ.
ϕ
B
C
Worksheet: Introduction to Trigonometry
Introduction to Trigonometry Worksheet
1) Calculate the following sine ratios. Round to four decimal places.
a) sin 15 °
c) sin 80 °
b) sin 60°
d) sin 100 °
2) Calculate the value the angle to the nearest degree.
a) sin
0.5879
c) sin
b) sin
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0.9994
d) sin
0.2635
0.4569
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3) For each triangle below, name:
(i) the hypotenuse
(ii) the side opposite the angle marked θ
(iii) the side adjacent the angle marked θ.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
Answers: 1) a) 0.2588 b) 0.8660 c) 0.9848 d) 0.9848 2) a) 36° b) 88° c)15° d) 27°
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A
C. Three types of Trigonometry Questions:
1) Solving for the angle
Solve for A:
sin A 
11.2 ft
6.4 ft
opp
hyp
Steps:
B
7.3 ft
C
1) Label triangle according to the angle of interest (in this case A) – hyp, opp, and adj.
2) Write down required formula (1 mark).
3) Fill in known values and leave unknown value (1 mark).
4) Solve. For the angle you must use the inverse function of sine. (1 mark)
Practice:
a)
b)
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A
2) Solving for the top of the ratio
sin C 
opp
hyp
4.3 cm
c
46º
Steps:
B
C
1) Label triangle according to the angle of interest (in this case C) – hyp, opp, and adj.
2) Write down required formula (1 mark).
3) Fill in known values and leave unknown value (1 mark).
4) Solve. (1 mark)
Practice:
a)
b)
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3) Solving for the bottom of the ratio
sin C 
opp
hyp
Steps:
1) Label triangle according to the angle of interest (in this case C) – hyp, opp, and adj.
2) Write down required formula (1 mark).
3) Fill in known values and leave unknown value (1 mark).
4) Solve. (1 mark)
Practice:
a)
b)
Worksheet: The Sine Ratio
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The Sine Ratio Worksheet
1) Use your calculator to determine the value of each of the following sine ratios to four
decimal places.
a) sin 30°
b) sin 48°
c) sin 62°
d) sin 77°
2) Calculate the angle to the nearest degree.
a) sin D = 0.5491
b) sin H = 0.9998
3) Solve for the indicated angle in the following diagrams.
a)
b)
c)
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d)
e)
4) Find the opposite side in the following diagrams. Round answers to one decimal
place.
a)
b)
c)
d)
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5) Find the length of the hypotenuse, to one decimal place, in the following diagrams.
a)
b)
c)
d)
6) A rafter makes an angle of 28° with the horizontal. If the rafter is 15 feet long, what
is the height at the rafter’s peak? Draw a diagram.
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7) How high is a weather balloon that is tied to the ground if it is attached to a 15 metre
string and the angle between the string and the ground is 35°? Draw a diagram.
8) How long is a guy wire that is attached 4.2 metres up a pole if it makes an angle of
52° with the ground? Draw a diagram.
9) A boat is carried with the current at an angle of 43° to the shore. If the river is
approximately 15 metres wide, how far does the boat travel before reaching the
opposite shore? Draw a diagram.
Answers:
1) a) 0.5000 b) 0.7431 c) 0.8829 d) 0.9744 2) a) 33° b) 89° 3) a) 38.7° b) 39.8°
c) 43.4° d) 32.4° e) 50° 4) a) 10.8 cm b) 11.3 cm c) 2.8 mm d) 8.7 km 5) a) 199.8 mm
b) 8.6 m c)154.7 mm d) 25.6 mm 6) 7.04 ft 7) 8.6m 8) 5.3 m 9) 22m
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D. Angle of Elevation and Depression
Trigonometry is useful in many real life situations. Two common types of angles are the
angles of elevation and depression.
When you look up the _____________
_____________________ is the angle formed between the horizontal and your line of
sight.
When you look down the ___________
____________________ is the angle formed between the horizontal and your line of
sight.
Example: You are flying a kite overhead. The angle of elevation is 65º. The length of
string used is 75 ft. How high is the kite?
Angle of Elevation and Depression Worksheet
1) George is in a hot air balloon that is 125 metres high. The angle of elevation from a
house below, to the balloon, is 18°. How far is George from the house?
2) The angle of elevation of a road is 4.5°. What is the length of the section of road if it
rises 16 metres?
3) The angle of elevation of a slide that is 3.6 metres long is 32°. How high above the
ground is the top of the slide?
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4) A ramp with a length of 21.2 metres has an angle of elevation of 15°. How high up
does it reach?
5) The angle of elevation from the bottom of a waterslide to the platform above is 20°. If
the waterslide is 25 metres long, how high is the platform?
6) A man walks at an angle of 68° north of east for 45 metres. How north of his starting
point is he?
Answers:
1) about 404.5 m 2) about 203.9 m
5) about 8.6 m 6) about 41.7 m
3) about 1.9 m
4) about 5.5 m
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