Exercise 8.1
Multiple Choice Questions (Q. No. 1-8)


Question. 3 The value of the expression cosec (75° + θ) – sec (15° – θ) – tan (55° + θ) + cot (35° – θ) is


Question. 5 If cos(a + b) = 0, then sin (a – b) can be reduced to
(a) cosb             (b) cos 2b              (c) sina             (d) sin 2a

Question. 6 The value of (tan l° tan 2° tan 3° … tan89°) is


Question. 8If ΔABC is right angled at C then the value of cos (A + B) is

Exercise 8.1
Multiple Choice Questions (Q. No. 9-15)
Question. 9 If sin A + sin² A = 1 then the value of (cos² A + cos⁴ A) is







⟹         tanθ = tan60°
⟹           θ = 60°
Hence, the Sun’s elevation is 60^°.
Exercise 8.2
Very Short Answer Type Questions (Q. No. 1-7)
Write whether True or False and justify your answer.

Question. 2 The value of the expression (cos² 23° – sin² 67°) is positive.
Solution. False

Question. 3 The value of the expression (sin 80° – cos 80°) is negative.
Solution. False
We know that for domain, 0° £ q £90° the value of sinq is increasing and that of cosq is decreasing .
∴        sin80° – cos80°> 0
So, the value of the given expression is positive.          

Question. 5 If cos A + cos² A = 1, then sin² A + sin⁴ A = l
Solution. True
Given, cosA + cos² A = 1
⟹     cos A = 1 – cos²A
⟹     cos A = sin² A              [∵ sin² A = 1 − cos²A]
⟹     cos² A = sin⁴ A             [Squaring both sides]
⟹     1 – sin² A = sin⁴ A         [∵ cos² A = 1 – sin² A]
⟹     sin² A + sin⁴ A = 1      
Question. 6 (tanθ + 2) (2 tanθ + 1) = 5 tanθ + sec²θ
Solution. False
LHS = (tanθ + 2)(2tanθ + 1)
        =2 tan²θ + 4 tanθ + tanθ + 2
        =2 (sec²θ -1) + 5 tanθ + 2         [∵ sec²θ -tan²θ = 1]
        =2sec²θ + 5tanθ ≠ RHS
Question. 7 If the length of the shadow of a tower is increasing, then the angle of elevation of the Sun is also increasing.
Solution. False
Let BC be the tower and DB = x be the length of initial shadow and DA = y be the the increase in length of shadow.
Also, let the angle of elevation of the sun changes from θ₁ to θ₂ with the increase in the length of shadow from x to (x + y).

Here, θ₁ is the exterior angle of ΔTSD
Therefore, θ₁ > θ₂.
Hence, the angle of elevation of the Sun is also increasing with an increase in the length of shadow.  
Exercise 8.2
Very Short Answer Type Questions (Q. No. 8-12)
Write whether True or False and justify your answer.
Question. 8 If a man standing on a platform 3m above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.
Solution. False
Let the observer stands on point P on a platform 3m above the surface of a lake. Also, let C be the cloud and C’ be its reflection in lake. Again, let height of the cloud from the surface of the platform be h and q₁ and q₂ be the angle of elevation and angle of deptression respectively of the cloud.




Question. 11 The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.
Solution. False


Question. 12 If the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains unchanged.
Solution. True

Case II: On increasing height of a tower and the distance of the point of observation from its foot, by 10%, we have:

Hence, the angle of elevation of the tower top remains unchanged.
Exercise 8.3
Short Answer Type Questions (Q. No. 1-7)
Prove the following questions:









Hence proved.
Exercise 8.3
Short Answer Type Questions (Q. No. 8-15)



Question. 10 A ladder 15m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then find the height of the wall.
Solution.
Let AB be the 15 m long ladder making a 60^o angle with a h metre high wall AC.

Question. 11 Simplify (1 + tan² θ) (1 – sin θ) (1 + sin θ).
Solution.

Question. 12 If 2sin²θ – cos²θ = 2, then find the value of θ.
Solution.


Question. 14 An observer 1.5m tall is 20.5m away from a tower 22m high. Determine the angle of elevation of the top of the tower from the eye of the observer.
Solution.
Let AB be the 22 m high tower and CD be the 1.5 m tall observer standing at a distance of BC = 20.5 m away from the tower.
Also, let θ be the angle of elevation of the top of the tower from the eye of the observer.

Question. 15 Show that tan⁴θ + tan² θ = sec⁴ θ – sec² θ.
Solution.

Exercise 8.4
Long Answer Type Questions (Q. No. 1-6)


Question. 3 The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20m towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower.
Solution.
Let AC be the h  metre high tower and B be the initial point from where angle of  elevation of the top of tower is observed.
Let after moving 20m towards tower, observer stands on point D with angle of elevation being 30^o +15^o = 45^o and x metres away from the foot of tower.



Question. 5 If sinθ  + 2cosθ = 1, then prove that 2 sinθ – cosθ = 2.
Solution.              

Question. 6 The angle of elevation of the top of a tower from two points distant s and t from its foot are

Solution.
Let AC be the h  metre high towerand θ and (90° – θ) be the angles of elevation for distances s and t respectively.
[As the two angles of elevation are given to be the complementary angles]


Exercise 8.4
Long Answer Type Questions (Q. No. 7-12)
Question. 7 The shadow of a tower standing on a level plane is found to be 50m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.
Solution:
Let AC be the tower of height h and60° and 30° be its angles of elevation at distances x m and (x + 50) m, respectively as shown below.


Question. 8 A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove



Hence Proved.
Question. 10 If sinθ + cosθ = p and secθ + cosecθ = q, then prove that q(p² -l)= 2p.
Solution:



Hence proved.