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.
Page No 5.19:
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:
Page No 5.40:
Question 4:
Prove that:
Answer:
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Question 5:
Prove that:
Answer:
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Question 6:
In a ∆ABC, prove that:
(i) cos (A + B) + cos C = 0
(ii)
(iii)
Answer:
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Question 7:
In a ∆A, 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:
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 = cos2 x, then write the value of cos2 x (1 + cos2 x).
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 + sin2 x = 1, then write the value of cos12 x + 3 cos10 x + 3 cos8 x + cos6 x.
Answer:
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Question 7:
If sin x + sin2 x = 1, then write the value of cos8 x + 2 cos6 x + cos4 x.
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.
Page No 5.41:
Question 11:
Write the value of 2 (sin6 x + cos6 x) −3 (sin4 x + cos4 x) + 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 tan4 A + cot4 A.
Answer:
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Question 15:
Write the least value of cos2 x + sec2 x.
Answer:
We know:
cos x can take the minimum value of .
cos2 x + sec2 x
Page No 5.41:
Question 16:
If x = sin14 x + 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 .
Page No 5.41:
Question 17:
If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.
Answer:
Page No 5.41:
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
Page No 5.41:
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 ϕ, y = r sin θ sin ϕ and z = r cos θ, then x2 + y2 + z2 is independent of
(a) θ, ϕ
(b) r, θ
(c) r, ϕ
(d) r.
Answer:
(a) θ, ϕ
We have:
x = r sin θ cos ϕ , y = r sin θ sin ϕ 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)
Page No 5.42:
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:
sin6 A + cos6 A + 3 sin2 A cos2 A =
(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 5.42:
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:
sin2 π/18 + sin2 π/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 tan4 A + cot4 A is equal to
(a) 110
(b) 191
(c) 80
(d) 194
Answer:
(d) 194
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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)
Page No 5.43:
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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