Page No 7.19:
Question 1:
If and , where 0 < A, , find the values of the following:
(i) sin (A + B)
(ii) cos (A + B)
(iii) sin (A − B)
(iv) cos (A − B)
Answer:
Now,
Page No 7.19:
Question 2:
(a) If , where < A < π and 0 < B < , find the following:
(i) sin (A + B)
(ii) cos (A + B)
(b) If , where A and B both lie in second quadrant, find the value of sin (A + B).
Answer:
Page No 7.19:
Question 3:
If , where π < A < < B < 2π, find the following:
(i) sin (A + B)
(ii) cos (A + B)
Answer:
Page No 7.19:
Question 4:
If , where π < A < and 0 < B < , find tan (A + B).
Answer:
Page No 7.19:
Question 5:
If , where < A < π and < B < 2π, find tan (A − B).
Answer:
Page No 7.19:
Question 6:
If , where < A < π and 0 < B < , find the following:
(i) tan (A + B)
(ii) tan (A − B)
Answer:
Page No 7.19:
Question 7:
Evaluate the following:
(i) sin 78° cos 18° − cos 78° sin 18°
(ii) cos 47° cos 13° − sin 47° sin 13°
(iii) sin 36° cos 9° + cos 36° sin 9°
(iv) cos 80° cos 20° + sin 80° sin 20°
Answer:
Page No 7.19:
Question 8:
If , where A lies in the second quadrant and B in the third quadrant, find the values of the following:
(i) sin (A + B)
(ii) cos (A + B)
(iii) tan (A + B)
Answer:
Page No 7.19:
Question 9:
Prove that:
Answer:
LHS = cos105o + cos15o
= cos(90o + 15o) + cos(90o 75o)
= - sin 15o + sin 75o [As cos(90o+A) = sin A and cos(90o B) = sin B]
= sin 75o sin 15o
= RHS
Hence proved.
Page No 7.19:
Question 10:
Prove that .
Answer:
Page No 7.19:
Question 11:
Prove that
(i) .
(ii)
(ii)
Answer:
(i)
Page No 7.19:
Question 12:
Prove that:
(i)
(ii)
(iii)
Answer:
(i)
(ii)
(iii)
Page No 7.19:
Question 13:
Prove that .
Answer:
Page No 7.20:
Question 14:
(i) If , prove that .
(ii) If , then prove that .
Answer:
(i)
(ii)
Page No 7.20:
Question 15:
Prove that:
(i)
(ii) sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
Answer:
(i)
Hence proved.
(ii)
Page No 7.20:
Question 16:
Prove that:
(i)
(ii)
(iii)
(iv) sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
(v) cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
(vi)
Answer:
Page No 7.20:
Question 17:
Prove that:
(i) tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
(ii)
(iii) tan 36° + tan 9° + tan 36° tan 9° = 1
(iv) tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
Answer:
Page No 7.20:
Question 18:
Prove that:
Answer:
Page No 7.20:
Question 19:
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
Answer:
Page No 7.20:
Question 20:
If tan A = x tan B, prove that .
Answer:
Page No 7.20:
Question 21:
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
Answer:
Page No 7.20:
Question 22:
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
Answer:
Page No 7.20:
Question 23:
If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) .
Answer:
Given:
Page No 7.20:
Question 24:
If x lies in the first quadrant and , then prove that:
Answer:
Page No 7.20:
Question 25:
If tan x + , then prove that .
Answer:
Page No 7.21:
Question 26:
If sin (α + β) = 1 and sin (α − β), where 0 ≤ α, , then find the values of tan (α + 2β) and tan (2α + β).
Answer:
Page No 7.21:
Question 27:
If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).
Answer:
Page No 7.21:
Question 28:
If sin α + sin β = a and cos α + cos β = b, show that
(i)
(ii)
Answer:
(i)
Now,
From (1) and (2), we have
(ii)
Page No 7.21:
Question 29:
Prove that:
(i)
(ii)
(iii)
Answer:
Page No 7.21:
Question 30:
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Answer:
Given:
Page No 7.21:
Question 31:
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Answer:
Page No 7.21:
Question 32:
If angle is divided into two parts such that the tangents of one part is times the tangent of other, and is their difference, then show that . [NCERT EXEMPLER]
Answer:
Let and be the two parts of angle . Then,
and (Given)
Now,
Applying componendo and dividendo, we get
Page No 7.21:
Question 33:
If , then show that . [NCERT EXEMPLER]
Answer:
Dividing numerator and denominator on the RHS by , we get
Now,
Page No 7.21:
Question 34:
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Answer:
Page No 7.26:
Question 1:
Find the maximum and minimum values of each of the following trigonometrical expressions:
(i) 12 sin x − 5 cos x
(ii) 12 cos x + 5 sin x + 4
(iii)
(iv) sin x − cos x + 1
Answer:
(i)
(ii)
(iii)
(iv)
Page No 7.26:
Question 2:
Reduce each of the following expressions to the sine and cosine of a single expression:
(i)
(ii) cos x − sin x
(iii) 24 cos x + 7 sin x
Answer:
Page No 7.26:
Question 3:
Show that sin 100° − sin 10° is positive.
Answer:
Page No 7.26:
Question 4:
Prove that lies between .
Answer:
Page No 7.27:
Question 1:
The value of is
(a)
(b)
(c) 1
(d) 0
Answer:
(b)
Page No 7.27:
Question 2:
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
(a) 0
(b) −1
(c) 1
(d) None of these
Answer:
(b) −1
π = 180°
We know that, ,
Now, using the identities and , we get
Page No 7.27:
Question 3:
tan 20° + tan 40° + tan 20° tan 40° is equal to
(a)
(b)
(c)
(d) 1
Answer:
(c)
Page No 7.27:
Question 4:
If , then the value of A + B is
(a) 0
(b)
(c)
(d)
Answer:
(d)
Page No 7.27:
Question 5:
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
(a) 0
(b) 5
(c) 1
(d) None of these
Answer:
(a) 0
Page No 7.27:
Question 6:
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
(a) 6
(b) 1
(c)
(d) None of these
Answer:
(c)
In triangle ABC,
If tan A+tan B+tan C =6,
tan A tan B tan C =6
Page No 7.27:
Question 7:
tan 3A − tan 2A − tan A =
(a) tan 3 A tan 2 A tan A
(b) −tan 3 A tan 2 A tan A
(c) tan A tan 2 A − tan 2 A tan 3 A − tan 3 A tan A
(d) None of these
Answer:
(a)
Page No 7.27:
Question 8:
If A + B + C = π, then is equal to
(a) tan A tan B tan C
(b) 0
(c) 1
(d) None of these
Answer:
(c) 1
π = 180°
Using tan(180 – A) = -tan A, we get:
Page No 7.27:
Question 9:
If , where P and Q both are acute angles. Then, the value of P − Q is
(a)
(b)
(c)
(d)
Answer:
(b) 60⁰ =
Hence, the correct answer is option B.
Page No 7.27:
Question 10:
If cot (α + β) = 0, sin (α + 2β) is equal to
(a) sin α
(b) cos 2 β
(c) cos α
(d) sin 2 α
Answer:
(a)
Page No 7.27:
Question 11:
(a) tan 55°
(b) cot 55°
(c) −tan 35°
(d) −cot 35°
Answer:
(a)
Page No 7.27:
Question 12:
The value of is
(a)
(b) 0
(c)
(d)
Answer:
(a)
Page No 7.27:
Question 13:
If tan θ1 tan θ2 = k, then
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 7.27:
Question 14:
If sin (π cos x) = cos (π sin x), then sin 2 x =
(a)
(b)
(c)
(d) none of these
Answer:
Page No 7.28:
Question 15:
If and , then the value of is
(a) (b) (c) 0 (d)
Answer:
It is given that and .
Now,
Hence, the correct answer is option D.
Page No 7.28:
Question 16:
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
(a) sin 2A
(b) cos 2A
(c) cos 3A
(d) sin 3A
Answer:
(b) cos 2A
Page No 7.28:
Question 17:
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
(a) a2 + 1
(b) a2 + 2
(c) a2 − 2
(d) None of these
Answer:
(c)
Page No 7.28:
Question 18:
If tan (A − B) = 1 and sec (A + B) = , the smallest positive value of B is
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 7.28:
Question 19:
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
(a) 2
(b) 1
(c) 0
(d) 3
Answer:
(a)
Page No 7.28:
Question 20:
The maximum value of is
(a) 1/2
(b) 3/2
(c) 1/4
(d) 3/4
Answer:
(b)
Page No 7.28:
Question 21:
If cos (A − B)and tan A tan B = 2, then
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 7.28:
Question 22:
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
(a) −1
(b)
(c)
(d) None of these
Answer:
(c)
Page No 7.28:
Question 23:
If then the value of (1 + tan α) (1 + tan β) is
(a) 1
(b) 2
(c) –2
(d) not defined
Answer:
Given
Page No 7.28:
Question 24:
The value of is
(a) 2 cosθ
(b) 2 sinθ
(c) 1
(d) 0
Answer:
Page No 7.28:
Question 25:
The minimum value of 3 cos x + 4 sin x + 8 is
(a) 5
(b) 9
(c) 7
(d) 3
Answer:
Page No 7.29:
Question 1:
The maximum value of 3 cos x + 4 sin x + 5 is _________.
Answer:
Page No 7.29:
Question 2:
The minimum value of 4 cos x – 3 sin x + 7 is _________.
Answer:
Page No 7.29:
Question 3:
If sinθ + cosθ = 1, then the value of sin 2θ is ___________.
Answer:
Page No 7.29:
Question 4:
If then (1 + tan A) (1 – tan B) = __________.
Answer:
Page No 7.29:
Question 5:
If then (1 + tan A) (1 + tan B) = ___________.
Answer:
Page No 7.29:
Question 6:
If and tan A tan B = 2, then sin A sin B = _______________.
Answer:
Page No 7.29:
Question 7:
If then x + y + z = ____________.
Answer:
Page No 7.29:
Question 8:
The value of is ___________.
Answer:
Page No 7.29:
Question 9:
If and 3 tan x = 4 tan y, then sin (x – y) is equal to __________.
Answer:
Page No 7.29:
Question 10:
If then k = _____________.
Answer:
Page No 7.29:
Question 11:
If and then the value of x + y is _____________.
Answer:
Page No 7.29:
Question 12:
The value of tan 5x tan 3x tan 2x – tan 5x + tan 3x + tan 2x is ____________.
Answer:
Page No 7.29:
Question 13:
The value of is ______________.
Answer:
Page No 7.29:
Question 1:
If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.
Answer:
Page No 7.29:
Question 2:
If x cos θ = y cos , then write the value of .
Answer:
Page No 7.29:
Question 3:
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
Answer:
Page No 7.29:
Question 4:
Write the maximum value of 12 sin x − 9 sin2 x.
Answer:
Page No 7.29:
Question 5:
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
Answer:
Page No 7.29:
Question 6:
Write the interval in which the value of 5 cos x + 3 cos lies.
Answer:
Page No 7.30:
Question 7:
If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.
Answer:
Page No 7.30:
Question 8:
If , then write the value of tan x tan y.
Answer:
Page No 7.30:
Question 9:
If a = b , then write the value of ab + bc + ca.
Answer:
Page No 7.30:
Question 10:
If A + B = C, then write the value of tan A tan B tan C.
Answer:
Page No 7.30:
Question 11:
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
Answer:
Page No 7.30:
Question 12:
If tan and , then write the value of α + β lying in the interval .
Answer:
View NCERT Solutions for all chapters of Class 12