MBF3C Angle of Elevation and Angle of Depression

Ms. Turk

Definition: Elevation usually means up. Depression usually means down.

This is an angle of elevation: This is an angle of depression:

In an angle of elevation, a line In an angle of depression, a line

rises or goes up from the horizontal. goes down from the horizontal

The angle of elevation shown here is 36.02°. / The angle of depression shown here is 54.14°

7) Draw an example an angle of elevation 8) Draw an example of an angle of depression

In the real world, trigonometry is used to solve for the height of very tall buildings or mountains, or the distance of a faraway object. Surveyors use a measuring instrument called a transit to help them find angles of depression or elevation.

1 A surveyor is standing on a cliff that is 80 m high. He is using a transit that is

1.5 m high. The surveyor sees a car parked at the bottom of the cliff. The angle of depression to the car is 34°. How far away is the car from the base of the cliff? Hint: Always draw a picture!

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From a point 10 m from a building, a surveyor measures the angle of elevation to the top of the building to be 65°. She is using a 1.5 m tall transit. How tall is the building?

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Kim and Yuri live in apartment buildings that are 30 m apart. The angle of depression from Kim’s balcony to where Yuri’s building meets the ground is 40°. The angle of elevation from Kim’s balcony to Yuri’s balcony is 20°.

a)  How high is Kim’s balcony above the ground, to the nearest metre?

b)  How high is Yuri’s balcony to the ground to the nearest metre?

1) Always label the triangles first: opposite, adjacent, and hypotenuse

2) Figure out which ratio you need using

3) Solve for the unknown angle or side

Adjacent is the side that touches the angle. Opposite is the side that does not touch the angle.

*Note: The opposite and adjacent sides depend on the angle! If we look at the same triangle but a different angle, the opposite and adjacent sides will be different, but the Hypotenuse always stays the same!

Identify the adjacent, opposite and hypotenuse sides for the angles in these right triangles:

Trigonometric Ratios

The following triangles are similar.

What do we know about the relationship between the sides of these triangles?

1 For the triangle below, find

/ a) length of hypotenuse side
b) length of adjacent side
c) length of opposite side

d) sin x e) cos x f) tan x

2 Calculate to 4 decimal places using your calculator.

*A note on Calculators* Make sure that your calculator is in Degrees, not Radians, or all your numbers will be wrong!

c)  sin 55°
d)  cos 34°
e)  tan 15° / f)  sin 115°
g)  cos 90°
h)  tan 96°

Using Trigonometric Ratios to Find Sides

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/ 1) First label the sides opposite, adjacent, hypotenuse.
2) Then look at the side we have, and the side we want to find.
3) Check for which ratio has Adjacent and Hypotenuse.

4 Find the side x.

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Using Trigonometric Ratios to find Angles

6 Calculate the angle x to the nearest degree

Now, we want x alone. To do this, we usually perform the opposite operation on x.

For example when we have 2x = 6,

2 is multiplying x, so we divide both sides by 2 to find x.

The opposite operation for sin is sin-1.

a)  sin x = 0.707
b)  cos x = 0.259
c)  tan x = 1.732 / d)  sin x = 0.848
e)  cos x = 0.985
f)  tan x = -5.671

Finding angle a.

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