Hwa Chong Institution
High School Section
Secondary Mathematics 2011
Name: _______________ Class: 2I3 ( )
Trigonometry Worksheet: Elevation and Depression
Date: ______________
About Trigonometry
1. Formed from Greek words 'trigonon' (triangle) and 'metron' (measure).
2. Trigonometric triangles are always right-angled triangles.
3. A branch of mathematics that studies triangles and relationships between their sides and the angles between sides.
4. Defines the trigonometric functions, which describe those relationships of triangles and is also applicable to our daily lives.
The three sides of a right-angled triangle
1. Hypotenuse- side opposite the right-angle, and longest side of triangle
2. Opposite- The side that is not attached to the theta of the triangle.
3. Adjacent- shares common side and a common vertex with the theta, but is not the longest side of the triangle.
Theta
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The eighth letter of the Greek alphabet.
A variable, not a constant.
It is the variable commonly used in trigonometry to represent an angle value.
Represented by the symbol ' θ '.
Prepared by Low Wei Yang, Lee Liak Ghee, Chew Tian-le, Ng Shen Han, 2011

Trigonometric Functions
1. Sin (Sine) = ratio of opposite side to the hypotenuse.
2. Cos (Cosine) = ratio of adjacent side to the hypotenuse.
3. Tan (Tangent) = ratio of opposite side to the adjacent side.
4. Cosec (Cosecant) = reciprocal of Sin
5.
Sec (Secant) = reciprocal of Cos
6.
Cotan (Cotangent) = reciprocal of Tan
Easier way to remember the three functions, Sin, Cos, and Tan.
TOA CAH SOH (Big foot auntie in Hokkien)
1. TOA: Tangent = Opposite ÷ Adjacent (T=O/A)
2. CAH: Cosine = Adjacent ÷ Hypotenuse (C=A/H)
3. SOH: Sine = Opposite ÷ Hypotenuse (S=O/H)
Angle of Elevation
The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight). This means that the observer is looking upwards, as his or her line of sight is elevated.
Prepared by Low Wei Yang, Lee Liak Ghee, Chew Tian-le, Ng Shen Han, 2011

Angle of Depression
If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression . This means that the observer is looking downwards, as his or her line of sight is depressed.
Applicability of Angles of Elevation and Depression
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Used by architects to design buildings by setting dimensions
Used by astronomers for locating apparent positions of celestial objects
Used in computer graphics by designing 3D effects properly
Used in nautical navigations by sailors (sextants)
Many other uses in our daily lives
Simple Word Problem involving Angles of Elevation
Little Tom, who is 0.75 metres tall is looking at a bug on the top of a big wall, which is 11 times his height. He is standing 2 metres away from the wall. What angle is he looking up at?
Solution:
1. Actual height of ceiling: 0.75m x (11)= 8.25m
2. Subtract off his own body height: 8.25m - 0.75m = 7.5m
3. tan( θ ) = 7.5m ÷ 2m
4. tan -1 (7.5 ÷ 2) = 75.1... o
Prepared by Low Wei Yang, Lee Liak Ghee, Chew Tian-le, Ng Shen Han, 2011

Simple Word Problem involving Angles of Depression
A boy 1m tall is standing on top of a staircase 33m high while looking at a patch of grass on the ground 50m away from him. Find the angle from which he is looking at.
Solution:
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Actual height boy is looking from: 33m + 1m = 34m sin( θ) = 34m ÷ 50m sin -1 (34 ÷ 50) = 42.8...
o
Overall summary
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To find the angle of elevation, we make use of tan -1 , also known as arc tangent. This allows us to find the angle once tan( θ ) is known, which is the ratio of the height to the adjacent side.
To find the angle of depression, we make use of sin -1 , also known arc sine.
This allows us to find the angle once sin( θ ) is known, which is the ratio of the opposite side of the angle to the hypotenuse.
Prepared by Low Wei Yang, Lee Liak Ghee, Chew Tian-le, Ng Shen Han, 2011

A few word problems to be solved:
1a) Chris, standing on the horizontal ground 40m from the base of a cliff and noted that the height of the cliff was 70m. What was the angle at which he was looking up? Assume that Chris’s eye level is 1 metre from his feet
1b) Chris, now standing on the top of the cliff now sees the ocean 180m from where he is standing. What is the angle at which he is looking down?
2a) The pilot of a small airplane flying 300m above the ground notices a village off the left wing 700m away from him. What is the angle at which he is looking down?
2b) When the pilot descends to 150m, he sees another plane fly above and perpendicular to him 400m away. What is the angle at which he is looking up?
Prepared by Low Wei Yang, Lee Liak Ghee, Chew Tian-le, Ng Shen Han, 2011

3) A tree 7 metres in height casts a shadow of length 9 metres. What is the angle of elevation from the end of the shadow to the top of the tree with respect to the ground?
4) An airplane is flying at a height of 2 miles above the runway. The distance along the ground from the airplane to the runway is 5 miles. What is the angle of depression from the airplane to the runway?
5) A building is 50 metres high. At a distance away from the building, an observer notices that the angle of elevation to the top of the building is 41º. How far is the observer from the base of the building?
6) A bird sits on top of a lamp-post. The angle of depression from the bird to the feet of an observer standing away from the lamppost is 35º. The distance from the bird to the observer is 25 meters. How tall is the lamp-post?
Prepared by Low Wei Yang, Lee Liak Ghee, Chew Tian-le, Ng Shen Han, 2011