Trigonometry & Height and Distances
Question 1 |
What is the maximum value of 3 Sinθ + 4 cosθ?
12 | |
5 | |
6 | |
1 |
Discuss it
Maximum value = √(a2 + b2) = 5
Question 2 |
What is minimum value of Sinθ + cosθ ?
-2 | |
√3/2 | |
-1 | |
-√2 |
Discuss it
Minimum value = √(a2 - b2)
=-√2
Question 3 |
If tan (x+y) tan (x-y) = 1, then find tan (2x/3)?
1/√3 | |
1/2 | |
1/√2 | |
2/√3 |
Discuss it
tanA = cotB,
tanA*tan B = 1
So, A +B = 90o
(x+y)+(x-y) = 90o, 2x = 90o , x = 45o
Tan (2x/3) = tan 30o = 1/√3
Question 4 |
Find the Value of tan30o + tan120o?
√3 | |
-1/√3 | |
0 | |
-2/√3 |
Discuss it
=tan(30) + tan(180-120) =1/√3 +(-√3) = -2/√3
Question 5 |
1/3 | |
4/3 | |
2 | |
3/4 |
Discuss it
=2(sin2θ + cos2θ) + cos2θ ; (by putting sin2θ + cos2=1)
= 2 + cos2θ ;(the minimum value of cos2θ=0) = 2 + 0 = 2
Question 6 |
The angle of elevation of the sun, when the length of the shadow of a tree is 1/√3 times the height of the tree, is:
30 degrees | |
45 degrees | |
60 degrees | |
90 degrees |
Discuss it
Let AB be the height of the tree and AC be the length of the shadow. We need to calculate the angle ACB where BC is the hypotenuse. Tanθ = Perpendicular / Base = 1/(1/√3)
Tanθ = Tan 60°
Therefore angle of elevation is 60 deg
Question 7 |
From a point A on a level ground, the angle of elevation of the top of a tower is 30 degrees. If the tower is 100 m high, the distance of point A from the foot of the tower is:
148 m | |
156 m | |
173 m | |
200 m |
Discuss it
Let QR be the tower. Then, QR = 100 m and angle QAR = 30 degrees. We know, cot 30° = √3 = AQ/QR. Therefore, AQ = 100*√3 = 173 m.
Question 8 |
Amit is standing at a point P is watching the top of a tower, which makes an angle of elevation of 45° with Amit's eye. He walks some distance towards the tower to watch its top and the angle of elevation becomes 60°. What is the distance between the base of the tower and the point P?
4.2 units | |
8 units | |
10 units | |
Data inadequate |
Discuss it
Let MN be the tower and Amit be standing at P (45° = angle MPN) and Q (60° = angle MQN). We are only given two angles and no sides of the triangles. Therefore, the data is inadequate.
Question 9 |
32 kmph | |
36 kmph | |
40 kmph | |
44 kmph |
Discuss it
Question 10 |
What is the area of the circle which has the diagonal of the square as its diameter if the area of square is ' d ' ?
πd | |
πd2 | |
(1/4) πd2 | |
(1/2)πd |
Discuss it
One important observation to solve the question : Diagonal of Square = Diameter of Circle. Let side of square be x. From Pythagoras theorem. Diagonal = √(2*x*x) We know area of square = x * x = d Diameter = Diagonal = √(2*d) Radius = √(d/2) Area of Circle = π * √(d/2) * √(d/2) = 1/2 * π * d