Page No 5.18:
Question 1:
stion Prove the following identities (1-16)
sec4 x− sec2 x = tan4 x + tan2 x
Answer:
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Question 2:
Prove the following identities (1-16)
Answer:
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Question 3:
Prove the following identities (1-16)
Answer:
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Question 4:
Prove the following identities (1-16)
Answer:
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Question 5:
Prove the following identities (1-16)
Answer:
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Question 6:
Prove the following identities (1-16)
Answer:
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Question 7:
Prove the following identities (1-16)
Answer:
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Question 8:
Prove the following identities (1-16)
Answer:
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Question 9:
Prove the following identities (1-16)
Answer:
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Question 10:
Prove the following identities (1-16)
Answer:
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Question 11:
Prove the following identities (1-16)
Answer:
= RHS
Hence proved.
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Question 12:
Prove the following identities (1-16)
Answer:
= RHS
Hence proved.
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Question 13:
Prove the following identities (1-17)
Answer:
Hence proved.
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Question 14:
Prove the following identities (1-16)
Answer:
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Question 15:
Prove the following identities (1-16)
Answer:
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Question 16:
Prove the following identities (1-16)
Answer:
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Question 17:
If , then prove that is also equal to a.
Answer:
Disclaimer: There is some error in the given question.
The question should have been
Question: If , then prove that is also equal to a.
So, the solution is done accordingly.
Solution:
Hence proved.
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Question 18:
If , then the values of tan x, sec x and cosec x
Answer:
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Question 19:
If , then find the values of .
Answer:
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Question 20:
If show that .
Answer:
Hence proved.
Page No 5.19:
Question 21:
If , then prove that .
Answer:
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Question 22:
If prove that .
Answer:
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Question 23:
If , then prove that , where
Answer:
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Question 24:
If , then shown that .
Answer:
Hence proved.
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Question 25:
Prove the:
Answer:
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Question 26:
If , prove that
(i)
(ii)
(iii)
Answer:
(i) LHS:
RHS:
LHS = RHS
Hence proved.
(ii) LHS:
Hence proved.
(iii) LHS:
Page No 5.25:
Question 1:
Find the values of the other five trigonometric functions in each of the following:
(i) x in quadrant III
(ii) x in quadrant II
(iii) x in quadrant III
(iv) x in quadrant I
Answer:
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Question 2:
If sin and x lies in the second quadrant, find the value of sec x + tan x.
Answer:
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Question 3:
If sin find the value of 8 tan .
Answer:
We have:
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Question 4:
If sin x + cos x = 0 and x lies in the fourth quadrant, find sin x and cos x.
Answer:
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Question 5:
If find the values of other five trigonometric functions and hence evaluate .
Answer:
Page No 5.39:
Question 1:
Find the values of the following trigonometric ratios:
(i)
(ii) sin 17π
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
Answer:
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Question 2:
Prove that:
(i) tan 225° cot 405° + tan 765° cot 675° = 0
(ii)
(iii) cos 24° + cos 55° + cos 125° + cos 204° + cos 300° =
(iv) tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
(v) cos 570° sin 510° + sin (−330°) cos (−390°) = 0
(vi)
(vii)
Answer:
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Question 3:
Prove that
(i)
(ii)
(iii)
(iv)
(v)
Answer:
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Question 4:
Prove that:
Answer:
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Question 5:
Prove that:
Answer:
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Question 6:
In a âABC, prove that:
(i) cos (A + B) + cos C = 0
(ii)
(iii)
Answer:
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Question 7:
In a âA, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that
cos(180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0
Answer:
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Question 8:
Find x from the following equations:
Answer:
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Question 9:
Prove that:
(i)
(ii)
(iii)
(iv)
(v)
Answer:
Page No 5.40:
Question 1:
If tan x = , then sec x − tan x is equal to
(a)
(b)
(c) 2x
(d)
Answer:
(a)
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Question 2:
If sec , then sec x + tan x =
(a)
(b)
(c)
(d)
Answer:
(b)
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Question 3:
If is equal to
(a) sec x − tan x
(b) sec x + tan x
(c) tan x − sec x
(d) none of these
Answer:
(c) tan x − sec x
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Question 4:
If π < x <2π, then is equal to
(a) cosec x + cot x
(b) cosec x − cot x
(c) −cosec x + cot x
(d) −cosec x − cot x
Answer:
(d) −cosec x − cot x
Page No 5.41:
Question 5:
If , and if , then y is equal to
(a)
(b)
(c)
(d)
Answer:
(b)
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Question 6:
If is equal to
(a) 2 sec x
(b) −2 sec x
(c) sec x
(d) −sec x
Answer:
(b) −2 sec x
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Question 7:
If x = r sin θ cos Ï, y = r sin θ sin Ï and z = r cos θ, then x2 + y2 + z2 is independent of
(a) θ, Ï
(b) r, θ
(c) r, Ï
(d) r.
Answer:
(a) θ, Ï
We have:
x = r sin θ cos Ï , y = r sin θ sin Ï and z = r cos θ,
∴ x2 + y2 + z2
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Question 8:
If tan x + sec x = , 0 < x < π, then x is equal to
(a)
(b)
(c)
(d)
Answer:
(c)
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Question 9:
If tan and θ lies in the IV quadrant, then the value of cos x is
(a)
(b)
(c)
(d)
Answer:
(a)
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Question 10:
If is equal to
(a) 1 − cot α
(b) 1 + cot α
(c) −1 + cot α
(d) −1 −cot α
Answer:
(d) −1 −cot α
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Question 11:
sin6A + cos6A + 3 sin2A cos2A =
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(b) 1
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Question 12:
If , then cos x is equal to
(a)
(b)
(c)
(d)
Answer:
(b)
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Question 13:
If , then tan x =
(a)
(b)
(c)
(d)
Answer:
(c)
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Question 14:
is true if and only if
(a)
x +
y ≠ 0
(b)
x =
y,
x ≠ 0
(c)
x =
y
(d)
x ≠0,
y ≠ 0
Answer:
(b) x = y, x ≠ 0
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Question 15:
If x is an acute angle and , then the value of is
(a) 3/4
(b) 1/2
(c) 2
(d) 5/4
Answer:
(a) 3/4
Page No 5.41:
Question 16:
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
(a) 7
(b) 8
(c) 9.5
(d) 10
Answer:
(c) 9.5
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Question 17:
sin2π/18 + sin2π/9 + sin2 7π/18 + sin2 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0
Answer:
(c) 2
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Question 18:
If tan A + cot A = 4, then tan4A + cot4A is equal to
(a) 110
(b) 191
(c) 80
(d) 194
Answer:
(d) 194
Page No 5.42:
Question 19:
If x sin 45° cos2 60° = , then x =
(a) 2
(b) 4
(c) 8
(d) 16
Answer:
(c) 8
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Question 20:
If A lies in second quadrant 3tanA + 4 = 0, then the value of 2cotA − 5cosA + sinA is equal to
(a) (b) (c) (d)
Answer:
It is given that .
Now,
Also,
So,
Hence, the correct answer is option B.
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Question 21:
If , then tan x =
(a)
(b)
(c)
(d)
Answer:
(c)
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Question 22:
If tan θ + sec θ =ex, then cos θ equals
(a)
(b)
(c)
(d)
Answer:
(b)
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Question 23:
If sec x + tan x = k, cos x =
(a)
(b)
(c)
(d)
Answer:
(b)
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Question 24:
If , then
(a) f(x) < 1 (b) f(x) = 1 (c) 1 < f(x) < 2 (d) f(x) ≥ 2
Answer:
Hence, the correct option is answer D.
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Question 25:
Which of the following is incorrect?
(a) (b) cos x = 1 (c) (d) tan x = 20
Answer:
(a) is correct as
(b) cos x = 1 is correct as
(c) is not correct as
(d) tan x = 20 is correct as tan x can take any real value.
Hence, the correct answer is option C.
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Question 26:
The value of is
(a) (b) 0 (c) 1 (d)
Answer:
Hence, the correct answer is option B.
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Question 27:
The value of is
(a) 0 (b) 1 (c) (d) not defined
Answer:
We know that,
So,
Hence, the correct answer is option B.
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Question 28:
Which of the following is correct?
(a) (b) (c) (d)
Answer:
We know that, 1 radian is approximately 57º.
Also, the value of sinx is always increasing for ( or sinx is an increasing function for ).
Now,
Hence, the correct answer is option B.
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Question 29:
If then sin θ is equal to
(a) but not
(b) or
(c) but not
(d) none of these
Answer:
Given
Since tan is negative in 2nd or 4th quadrant
⇒ θ lies in 2nd or 4th quadrant
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Since AC2 = AB2 + BC2 = (–4)2 + (3)2
AC2 = 25
AC = ± 5
i.e. AC = 5
Hence, the correct answer is option B.
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Question 30:
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation
(a) a2 + b2 + 2ac = 0
(b) a2 – b2 + 2ac = 0
(c) a2 + c2 + 2ab = 0
(d) a2 – b2 – 2ac = 0
Answer:
Given sinθ and cosθ are roots of ax2 – bx + c = 0
Sum of roots is and product of root is
i.e
Since sin2θ + cos2θ = 1
Hence, the correct answer is option B.
Page No 5.43:
Question 31:
If sin θ + cosec θ = 2, then sin2θ + cosec2θ is equal to
(a) 1
(b) 4
(c) 2
(d) none of these
Answer:
Given sinθ + cosecθ = 2
⇒ (sinθ + cosecθ)2 = 4
i.e sin2θ + cosec2θ + 2 sinθ cosecθ = 4
Hence, the correct answer is option C.
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Question 32:
Which of the following is incorrect?
(a)
(b) cosθ = 1
(c)
(d) tanθ = 20
Answer:
i.e No solution
Hence, the correct answer is option C.
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Question 33:
If for real values of then
(a) θ is an acute angle
(b) θ is a right angle
(c) θ is an obtuse angle
(d) No value of θ is possible
Answer:
Given for real value of x,
which is not possible (âµ –1 ≤ cosθ ≤ 1)
∴ No such value of θ is possible
Hence, the correct answer is option D.
Page No 5.43:
Question 1:
The value of is _____________.
Answer:
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Question 2:
The value of is _____________.
Answer:
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Question 3:
The values of lie in the interval __________ .
Answer:
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Question 4:
If sin x + cos x = a, then sin x – cos x = __________.
Answer:
Given sin x + cos x = a
i.e (sin x – cos x)2 =a2 (squaring both side)
Now, consider
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Question 5:
If then is equal to _________.
Answer:
If is given,
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Question 6:
If then sec x – tan x = ___________.
Answer:
Given
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Question 7:
If then tan x = ___________.
Answer:
Subtracting (1) from (2)
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Question 8:
If cosec x + cot x = α, then sin x = _________.
Answer:
Given cosec x + cot x = a (1)
Since cosec2x – cot2x = 1
(cosec x – cot x) (cosec x + cot x) = 1
i.e (cosec x – cot x) a = 1
adding (1) and (2)
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Question 9:
If then the value of tan x is ______________.
Answer:
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Question 10:
If then the value of tan x is __________ .
Answer:
Given i.e x lies in III or IV quadrant
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Question 11:
If sin x + cosec x = 2, then sin2x + cosec2x = ___________ .
Answer:
Given sin x + cosec x = 2
Squaring both sides, (sin x + cosec x)2 = 4
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Question 12:
The value of tan is ______________ .
Answer:
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Question 13:
The value of 3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x) is ___________.
Answer:
3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x)
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Question 14:
Given x > 0, the value of lie in the interval ____________.
Answer:
Given x > 0,
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Question 15:
If sin x + cos x = a, then sin6x + cos6x = __________.
Answer:
Given sin x + cos x = a
Squaring both sides, (sin x + cos x)2 = a2
Using identity, we have
Hence, sin6x + cos6x
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Question 16:
The value of cos 1° cos 2° cos 3° _______cos 179° is ____________.
Answer:
cos 1° cos 2° cos 3° ....... cos 179°
= cos 1° cos 2° cos 3° ......... cos 90° cos 91° ......... cos 179°
= cos 1° × cos 2° × cos 3° ..... × 0 × cos 91° .......... cos 179° (âµ cos 90° = 0)
= 0 × finite
= 0
Hence, cos 1° cos 2° cos 3° ......... cos 179° = 0
Page No 5.43:
Question 17:
The value of tan 1° tan 2° tan 3° _________ tan 89° is _________.
Answer:
tan 1° tan 2° tan 3° ........ tan 89°
Since 89° = 90° – 1
88° = 90° – 2 ........... 46° = 90° – 44°
and tan 89° = tan (90° – 1°) = cot 1°
tan 88° = tan (90° – 2°) = cot 2°
.
.
.
tan 46° = tan (90° – 44°) = cot 44°
tan 45° = 1
∴ tan 1° tan 2° tan 3° .......... tan 45° cot 44° .....cot 3° cot 2° cot 1°
= 1
Hence, tan 1° tan 2° ....... tan 89° = 1.
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Question 18:
If then k = ___________.
Answer:
If
Given
(correction)
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Question 19:
If then k = ___________.
Answer:
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Question 20:
If then k = ___________.
Answer:
If π < x < 2π
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Question 21:
The minimum value of 9 tan2θ + 4 cot2θ is ____________.
Answer:
9 tan2θ + 4 cot2θ
Since Arithmetic mass ≥ Geometric mean for 2 tans.
i.e minimum value of 9 tan2θ + 4 cot2θ is 12.
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Question 22:
If sec x = m and tan x = n, then is equal to ____________.
Answer:
If sec x = m and tan x = n
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Question 23:
If cos2x + sin x + 1 = 0, and 0 < x < 2π then x = _________.
Answer:
Given,
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Question 24:
If and x lies in the second quadrant, then cos x = ___________.
Answer:
Given and x lies in 2nd quadrant
Since sin2x + cos2x = 1
cos2x = 1 – sin2x
Since x lies in II Quadrant
⇒ cos x < 0
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Question 25:
If then the value of sec x + tan x is _________.
Answer:
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Question 26:
If tan x + cot x = 4, then tan4x + cot4x = ___________.
Answer:
Given tan x + cot x = 4
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Question 27:
If then is equal to ___________.
Answer:
Given
Page No 5.44:
Question 1:
Write the maximum and minimum values of cos (cos x).
Answer:
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Question 2:
Write the maximum and minimum values of sin (sin x).
Answer:
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Question 3:
Write the maximum value of sin (cos x).
Answer:
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Question 4:
If sin x = cos2x, then write the value of cos2x (1 + cos2x).
Answer:
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Question 5:
If sin x + cosec x = 2, then write the value of sinn x + cosecn x.
Answer:
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Question 6:
If sin x + sin2x = 1, then write the value of cos12x + 3 cos10x + 3 cos8x + cos6x.
Answer:
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Question 7:
If sin x + sin2x = 1, then write the value of cos8x + 2 cos6x + cos4x.
Answer:
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Question 8:
If sin θ1 + sin θ2 + sin θ3 = 3, then write the value of cos θ1+ cos θ2 + cos θ3.
Answer:
Sine function can take the maximum value of 1.
If, , then we have:
sin = 1
⇒ =
Similarly,
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Question 9:
Write the value of sin 10° + sin 20° + sin 30° + ... + sin 360°.
Answer:
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Question 10:
A circular wire of radius 15 cm is cut and bent so as to lie along the circumference of a loop of radius 120 cm. Write the measure of the angle subtended by it at the centre of the loop.
Answer:
Circumference of the circle of radius 15 cm:
Now, 94.2 cm will be the length of arc for the circle with radius 120 cm.
We know:
45 = radians
Therefore, the angle subtended by it at the centre of the loop is 45.
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Question 11:
Write the value of 2 (sin6x + cos6 x) −3 (sin4 x + cos4x) + 1.
Answer:
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Question 12:
Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.
Answer:
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Question 13:
If cot (α + β) = 0, then write the value of sin (α + 2β).
Answer:
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Question 14:
If tan A + cot A = 4, then write the value of tan4A + cot4A.
Answer:
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Question 15:
Write the least value of cos2x + sec2x.
Answer:
We know:
cos x can take the minimum value of .
cos2x + sec2x
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Question 16:
If x = sin14x + cos20 x, then write the smallest interval in which the value of x lie.
Answer:
If x = 0, 90, 180, 270, 360, then
The smallest interval in which the value of x lie is .
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Question 17:
If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.
Answer:
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