Pythagoras theorem
Definition of Trigonometric Ratio
Trigonometric ratios of commonly used angles
Angle of elevation
Angle of depression
The relationship between the hypotenuse, base and perpendicular in a right angled triangle is given by
h² = p² + b²
Definition of Trigonometric angles
1. Sine - Sine of an angle is the ratio of the height of a right angled triangle to the hypotenuse of a right
angled triangle.
or mathematically, Sin θ = p/h
2.Cosine - Cosine of an angle is the ratio of the base of a right angled triangle to the hypotenuse.
or mathematically, Cosθ = b/h
3.Tangent - Tangent of an angle is the ratio of the height of a right angled triangle to hypotenuse of a right
angled triangle ..
or mathematically, Tan θ = p/b
Trigonometric ratios of commonly used angles
| Angle (θ in degrees) | Sinθ | Cosθ | Tanθ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 30 | 1/2 | √3/2 | 1/√3 |
| 45 | 1/√2 | 1/√2 | 1 |
| 60 | √3/2 | 1/2 | √3 |
| 90 | 1 | 0 | Not Defined |
Angle of elevation
Angle of elevation as appeared in the figure, on the right, is the angle formed when looking from a horizontal platform to a point that is above the horizontal platform.
Angle of elevation is used to measure height of poles, mountains and measure distance between two objects if height of one is known.
Angle of depression
Angle of depression is the angle formed when looking down from a higher point to a point on a ground or on a horizontal surface.
Angle of elevation is used to measure distances or heights between two objects if one is known.
Solved Questions on Trigonometry
1. Two poles of equal heights are standing opposite to each other on either side of a road which is 100m wide. From a point between them on road, angles of elevation of their tops are 30° and 60°. The height ofeach pole in metre, is
(a) 25 √3 (b) 20 √3
(c) 28 √3 (d) 30 √3
Answer:(a) 25 √3
Solution:
Assume height of pole = p meter
Let the distance of the point from the first pole be (X) meter with angle of elevation 30° and then the distance from the second becomes (100-X) with angle of elevation 60° .
Then, tan30° = p/x
=> 1/√3 = p/x
=> x = p√3 ------ (i)
Again, for the second pole, we have
tan60°= p/(100-x)
=> √3 = p/(100-x)
=> p= √3 (100-x)
Using (i), we have
p = √3(100 -p√3)
p = 100√3 - 3p
4p = 100√3
p = 25√3
Short Cut:
Since both the poles are of equal length, then the distance between the two poles and the point is inversely proportional to the respective angles of elevation.
Therefore, (100-x)/x = tan30° /tan60° = (1/√3/√3/1)
=> (100-x)/x = 1/3
=> 300 - 3x = x = > 4x = 300 => x= 75 m
Let the height of pole be p meter. Then, tan60°= p/75
=> 75 √3 = p
=> p = 25√3
2.The shadow of a tower standing on a level ground is found to be 40 m longer when the sun's altitude is 30° than when it is 60°. Find the length of the tower.
(a) 10√3 (b) 20 m
(c) 20√3 (d) 10 m
Answer:20√3 = p
Solution: The distance of a point from the tower is inversely proportional to the tangent of the angle.
Let the height of the pole be p meter and the distance from the tower is x meter at 60° .
Thus, tan60°/tan30° = (x+40)/x
=> √3/1/√3 = (x+40)/x => (√3)(√3) = (x+40)/x
=> 3x = x+40 => 2x = 40
=> x = 20 meter
Using this to get
tan60° = p/20
=> 20√3 = p
3. The length of the shadow of the tower is 9 meters when the sun's altitude is 30°. What is the height of the tower?
(a) 9√3 m (b) 9√3/2 m
(c) 3√3 m (d) 4(1/2) m
Answer:(a) 9√3 m
Solution:
"when the sun's altitude is 30°" means the angle of the depression is 30°.
Therefore, the angle of elevation is 60°. Thus,
Tan 60° = p/9
=> 9√3 = p
4. The value of (secθ + cosecθ) when θ = 45°, is
(a) 4√2 (b) 2√2
(c) 5√2 (d) 3√2
Answer: (b) 2√2
Solution:
We know, sec 45° = √2 and cosec 45° = √2
Thus, (sec 45° + cosec 45°) = (√2 +√2) = 2√2
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