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Math 11AD
Right Angle Trigonometry
Ancient Mathematicians were able to find values for the ratios corresponding to any acute angle in a
right triangle, and it soon became apparent that they should name these ratios since they were so
commonly used.
The ratio of sine is opposite to hypotenuse
The ratio of cosine is adjacent to hypotenuse
The ratio of tangent is opposite to adjacent
Since each of these ratios depends on the measure of an acute angle  , they are often referred to as
functions of an acute angle and written in function form:
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
These trig ratios have an input of an ANGLE and their output is a RATIO.
Which of these triangles best represents . . .
A

B
C


a) a very small value of sin  ?
b) a very large value of tan  ?
c) a tan  of around 1?
d) a cos value that is very small?
e) a sin  value that is very large?
What does it mean when calculator spits out sin( 57)  0.8386705 ?
Inverse Trigonometry
So what if you know the ratio of the sides? Well then we can use the inverse trig functions to find the
angles of the triangle. The input of the inverse functions is a RATIO and their output is an ANGLE.
You have worked with these already, remember? They are the 2nd function of the primary trig buttons
on your calculator.
When you enter cos 1 (1.2) on your calculator, you get “error”.
Why?
Angle of Elevation and Depression
The angle of elevation is the angle through which your line of sight moves from the horizontal to a
target upwards. The angle of depression is the angle through which your line of sight moves from the
horizontal to a target downwards.
*** Note: The angle of
depression from the person to
the ant is the SAME as the
angle of elevation from the ant
to the person because of the ZTheorem! (The angle of
depression from the eagle to
the person is the same as the
angle of elevation from the
person to the eagle as well)
Right Angle Trigonometry Questions – round to the nearest hundredth unless otherwise indicated.
1.
Solve for the unknown side(s) in the following triangles:
a)
2.
b)
Solve for the unknown angles in the following triangles:
a)
3.
c)
b)
Q

650
T
R
U
P
In the diagram at left, T falls on line segment UP,
side TU = 10 cm and TP = 20cm. Determine the
size of angle θ.
Right ΔABC with 90° at C,
B = 53° and a = 7.5 cm.
For each of the following situations draw a diagram and solve.
4.
A campsite is 250 m from a cliff. When the campers awake in the morning they measure the
angle of elevation to the top of the cliff to be 40 . How tall is the cliff?
5.
Some dude is standing on the top of a 100 m tower. He looks down through an angle of
depression of 57 and sees a hotdog stand, nothing like some good street meat to satisfy a
craving! How far is the hotdog vendor from the base of the tower the dude is standing on?
6.
From her hotel room window on the sixth floor, Lily notices some window washers high above
her on the hotel across the street. Curious as to their height above ground, she quickly estimates
the two hotels are 50 m apart, the angle of elevation to the workers is 80 , and the angle of
depression to the base of the hotel is 50 .
a)
b)
7.
How high above the ground is the window in Lily’s room?
How high above the ground are the workers?
A sniper is lying down on the top of a 25 m platform under some serious camo. She looks up at
an angle of elevation of 16.7 to the hotel roof where her target is located. She looks down to
the bottom of that hotel at an angle of depression of 5.7 .
a)
b)
c)
How far away is the platform from the hotel?
How tall is the building?
How far will the bullet have to travel to hit her target?
8. On a beautiful day at the beach this summer, you took your most colourful kite to fly high in the sky.
You let out all 86 metres of string and the angle the sting makes with the ground is measured to be
36 degrees. How high in the sky is your kite?
9. A mountain road drops a vertical height of 32 metres for every 200 metres travelled along the road.
Truckers need to know the angle of inclination on steep hills in order to gear down. Calculate the
angle at which the road is inclined.
10. A lighthouse is 40 metres above sea level. From the top of the lighthouse the angle of depression to a
buoy is 35 degrees. How far is the buoy from the foot of the lighthouse?
11. Find the length of the diagonal of a rectangle if the diagonal makes an angle of 38 degrees with the
adjacent side, whose length is 15 cm.
12. A monument, 18 metres high, casts a shadow that is 22 metres long. Calculate the inclination of the
sun to the horizontal at this time of day.
13. One of Canada’s tallest trees is a Douglas fir on Vancouver Island. The angle of elevation measured
by an observer who is 78 metres from the base of the tree is 50 degrees. How tall is this tree to the
nearest metre?
14. Comfortable stairs have a slope of ¾. What angle do the stairs make with the horizontal, to the
nearest degree?
15. A coast guard patrol boat is 14.8 km east of the Brier Island lighthouse. A disabled yacht is 7.5 km
south of the lighthouse.
a) How far is the patrol boat from the yacht, to the nearest tenth of a kilometre?
b) At what angle south of due west, to the nearest degree, should the patrol boat travel to reach the
yacht?
Answers (in no particular order):
a) 16.18 cm
g)
59.59 m
m) 45.92

b) 60
c)

h) 44.08
n)
250.47 m
261.50 m
d) 19.04cm

e)
209.77 m
f)
100.14 m
k)
14.82 cm
l)
343.15 m
i) 64.74m
j) 30
o) 11.76 cm
p) 47.00
q) 9.95 cm
r) 12.46 cm
v) 16.6 km
w) 57.13 m
x) 50.55 m
s) 9.21˚
t) 39.29 ˚
u) 36.87 ˚
y) 93 m
z) 26.87 ˚
☼) 27 ˚