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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Medium

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Showing 61-90 of 157 questions
61 Medium
61 of 157
Application +4 | -1
If sin⁡(A+B+C)=1,tan⁡(A−B)=13\sin (A+B+C)=1, \tan (A-B)=\frac{1}{\sqrt{3}} and sec⁡(A+C)=2\sec (A+C)=2, then
View answer and explanation
Correct option: C
A=60∘,B=30∘,C=0∘A=60^{\circ}, B=30^{\circ}, C=0^{\circ}
Option C (A=60∘,B=30∘,C=0∘A=60^{\circ}, B=30^{\circ}, C=0^{\circ}) is correct because solving the system of equations derived from the given trigonometric relations yields these values for A, B, and C.
62 Medium
62 of 157
Understanding +4 | -1
cos⁡  1∘+cos⁡  2∘+cos⁡  3∘+...+cos⁡  180∘\cos \,\,1^\circ + \cos \,\,2^\circ + \cos \,\,3^\circ + ... + \cos \,\,180^\circ is equal to
View answer and explanation
Correct option: D
−1- 1
Option D (-1) is correct because the sum telescopes due to the symmetry of cosine around 90 degrees, with cos(180) being -1.
63 Medium
63 of 157
Application +4 | -1
If A=130∘A = 130^\circ and x=sin⁡A+cos⁡Ax = \sin A + \cos A, then
View answer and explanation
Correct option: A
x>0x > 0
Option A (x>0x > 0) is correct because sin⁡130∘+cos⁡130∘\sin 130^\circ + \cos 130^\circ simplifies to sin⁡50∘−sin⁡40∘\sin 50^\circ - \sin 40^\circ, which is positive since sin⁡50∘>sin⁡40∘\sin 50^\circ > \sin 40^\circ.
64 Medium
64 of 157
Application +4 | -1
If sin⁡A−6cos⁡A=7cos⁡A\sin A-\sqrt{6} \cos A=\sqrt{7} \cos A, then cos⁡A+6sin⁡A\cos A+\sqrt{6} \sin A is equal to
View answer and explanation
Correct option: B
7sin⁡A\sqrt{ } 7 \sin A
Option B (7sin⁡A\sqrt{7} \sin A) is correct because by manipulating the given equation, we can express cos⁡A\cos A in terms of sin⁡A\sin A and then substitute to find the value of cos⁡A+6sin⁡A\cos A + \sqrt{6} \sin A.
65 Medium
65 of 157
Application +4 | -1
The minimum value of sin⁡θcos⁡θ\sin \theta \cos \theta is
View answer and explanation
Correct option: C
−12-\frac{1}{2}
Option C (-12\frac{{1}}{{2}}) is correct because the minimum value of sin⁡θcos⁡θ\sin \theta \cos \theta is -12\frac{{1}}{{2}}, which can be found using trigonometric identities or calculus.
66 Medium
66 of 157
Application +4 | -1
The value of sin⁡47∘+sin⁡61∘−sin⁡11∘−sin⁡25∘=\sin 47^{\circ}+\sin 61^{\circ}-\sin 11^{\circ}-\sin 25^{\circ}=
View answer and explanation
Correct option: D
cos⁡7∘\cos 7^{\circ}
Option D (cos⁡7∘\cos 7^{\circ}) is the correct answer because after applying trigonometric identities, the expression simplifies to cos⁡7∘\cos 7^{\circ}.
67 Medium
67 of 157
Analyzing +4 | -1
The range of cos⁡θ(sin⁡θ+sin⁡2θ+sin⁡2α)\cos \theta\left(\sin \theta+\sqrt{\sin ^2 \theta+\sin ^2 \alpha}\right) is
View answer and explanation
Correct option: A
[−1+sin⁡2α,1+sin⁡2α]\left[-\sqrt{1+\sin ^2 \alpha}, \sqrt{1+\sin ^2 \alpha}\right]
Option A is correct because the range is found by solving the trigonometric equation and considering the discriminant of the resulting quadratic equation in terms of tan θ.
68 Medium
68 of 157
Application +4 | -1
If sin⁡2θ=x2+y2+12x\sin ^2 \theta=\frac{x^2+y^2+1}{2 x}, then xx must be
View answer and explanation
Correct option: C
1
Option C (1) is correct because the given equation implies that xx must be 1 to satisfy the condition 0≤sin⁡2θ≤10 \leq \sin^2 \theta \leq 1.
69 Medium
69 of 157
Understanding +4 | -1
How many times do the hands of the watch form an angle of 180 degree during a complete day?
View answer and explanation
Correct option: B
22 times
Option B (22 times) is correct because the hands of a watch form a 180-degree angle 22 times in a 24-hour period.
70 Medium
70 of 157
Application +4 | -1
The value of cos⁡480∘sin⁡150∘+sin⁡60∘⋅cos⁡390∘\cos 480^{\circ} \sin 150^{\circ}+\sin 60^{\circ} \cdot \cos 390^{\circ} is equal to
View answer and explanation
Correct option: C
1/21 / 2
Option C (1/21 / 2) is correct because substituting the values of the trigonometric functions and simplifying the expression leads to the result 1/21/2.
71 Medium
71 of 157
Analyzing +4 | -1
sec⁡  θ=a2+  b2a2−  b2\sec \,\,\theta = \frac{{{a^2} + \,\,{b^2}}}{{{a^2} - \,\,{b^2}}}, where a,   b, ∈Ra,\,\;b,\, \in R gives real values of θ\theta if and only if

View answer and explanation
Correct option: B
∣ a ∣≠∣ b ∣≠0\left| {\,a\,} \right| \ne \left| {\,b\,} \right| \ne 0
Option B (\left| {\,a\} \right| \ne \left| {\,b\} \right| \ne 0\left| {\,a\} \right| \ne \left| {\,b\} \right| \ne 0) is correct because it ensures that the value of sec⁡θ\sec \theta is real and either greater than or equal to 1, or less than or equal to -1.
72 Medium
72 of 157
Analyzing +4 | -1
If sin⁡  x+sin⁡2x=1\sin \,\,x + {\sin ^2}x = 1, then the value of cos⁡12  x+3  cos⁡10x+3  cos⁡8 x+cos⁡6  x+2  cos⁡4  x+cos⁡2x−2{\cos ^{12}}\,\,x + 3\,\,{\cos ^{10}}x + 3\,\,{\cos ^8}\,x + {\cos ^6}\,\,x + 2\,\,{\cos ^4}\,\,x + {\cos ^2}x - 2, is equal to
View answer and explanation
Correct option: D
sin⁡2  x{\sin ^2}\,\,x
Option D (sin⁡2  x{\sin ^2}\,\,x) is correct because the given expression simplifies to sin⁡2  x{\sin ^2}\,\,x using the trigonometric identity and the given condition.
73 Medium
73 of 157
Application +4 | -1
If xcos⁡θ+ysin⁡θ=zx \cos \theta+y \sin \theta=z, then what is the value of (xsin⁡θ−ycos⁡θ)2(x \sin \theta-y \cos \theta)^{2} ?
View answer and explanation
Correct option: A
x2+y2−z2x^{2}+y^{2}-z^{2}
Option A (x2+y2−z2x^{2}+y^{2}-z^{2}) is correct because it is derived by manipulating the given equation and using trigonometric identities.
74 Medium
74 of 157
Application +4 | -1
If cot⁡A=tan⁡((n−1)A),\cot A = \tan ((n - 1)A), where n∈N,n \in N, then a value of A can be
View answer and explanation
Correct option: A
π/2n\pi / 2 n
Option A (π/2n\pi / 2 n) is correct because substituting this value into the given equation satisfies the trigonometric relationship between cotangent and tangent.
75 Medium
75 of 157
Application +4 | -1
The value of 3  [sin⁡4(3π2−α)+sin⁡4(3π+α)]−2[sin⁡6(π2+α)+sin⁡6(5π−α)]  3\,\,\left[ {{{\sin }^4}\left( {\frac{{3\pi }}{2} - \alpha } \right) + {{\sin }^4}\left( {3\pi + \alpha } \right)} \right] - 2\left[ {{{\sin }^6}\left( {\frac{\pi }{2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]\; is equal to
View answer and explanation
Correct option: B
1
Option B (1) is correct because after simplifying the trigonometric expressions and applying trigonometric identities, the expression evaluates to 1.

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76 Medium
76 of 157
Application +4 | -1
Let f(θ)=cot⁡θ1+cot⁡θf(\theta)=\frac{\cot \theta}{1+\cot \theta} and α+β=5π4\alpha+\beta=\frac{5 \pi}{4}. Then the value of f(α)⋅f(β)f(\alpha) \cdot f(\beta) is
View answer and explanation
Correct option: C
1/21 / 2
Option C (1/21/2) is correct because by using the given condition α+β=5π4\alpha + \beta = \frac{{5\pi}}{{4}} and trigonometric identities, f(α)⋅f(β)f(\alpha) \cdot f(\beta) simplifies to 1/21/2.
77 Medium
77 of 157
Application +4 | -1
sin⁡2A+sin⁡2B−sin⁡2C=\sin ^2 A+\sin ^2 B-\sin ^2 C=
View answer and explanation
Correct option: D
2sin⁡Asin⁡Bcos⁡C2 \sin A \sin B \cos C
Option D (2sin⁡Asin⁡Bcos⁡C2 \sin A \sin B \cos C) is correct because by using trigonometric identities and sum-to-product formulas, the expression simplifies to the given form.
78 Medium
78 of 157
Application +4 | -1
The new position of the point (1,2)(1,2) under rotation through an angle of 90∘90^{\circ} about the origin in anticlockwise direction is
View answer and explanation
Correct option: A
(−2,1)(-2,1)
Option A ((-2,1)) is correct because it represents the new coordinates of the point (1,2) after a 90-degree anticlockwise rotation about the origin.
79 Medium
79 of 157
Application +4 | -1
The value of tan⁡A+tan⁡(60∘+A)−tan⁡(60∘−A)\tan \mathrm{A}+\tan \left(60^{\circ}+\mathrm{A}\right)-\tan \left(60^{\circ}-\mathrm{A}\right) is
View answer and explanation
Correct option: C
3tan⁡3A3 \tan 3 \mathrm{A}
Option C (3tan⁡3A3 \tan 3 \mathrm{A}) is correct because by using the tangent addition and subtraction formulas, the expression can be simplified to 3tan⁡3A3 \tan 3A.
80 Medium
80 of 157
Application +4 | -1
The value of 2(sin⁡6θ+cos⁡6θ)−3(sin⁡4θ+cos⁡4θ)+12\left(\sin ^6 \theta+\cos ^6 \theta\right)-3\left(\sin ^4 \theta+\cos ^4 \theta\right)+1 is
View answer and explanation
Correct option: B
0
Option B (0) is correct because by using trigonometric identities, the expression 2(sin⁡6θ+cos⁡6θ)−3(sin⁡4θ+cos⁡4θ)+12(\sin^6 \theta + \cos^6 \theta) - 3(\sin^4 \theta + \cos^4 \theta) + 1 simplifies to 0.
81 Medium
81 of 157
Application +4 | -1
If α,β,γ∈[0,π]\alpha, \beta, \gamma \in[0, \pi] and α,β,γ\alpha, \beta, \gamma are in AP, then sin⁡α−sin⁡γcos⁡γ−cos⁡α\frac{\sin \alpha-\sin \gamma}{\cos \gamma-\cos \alpha} is equal to
View answer and explanation
Correct option: C
cot⁡β\cot \beta
Option C (cot⁡β\cot \beta) is correct because, given that α,β,γ\alpha, \beta, \gamma are in AP, we can simplify the expression sin⁡α−sin⁡γcos⁡γ−cos⁡α\frac{{\sin \alpha - \sin \gamma}}{{\cos \gamma - \cos \alpha}} to cot⁡β\cot \beta.
82 Medium
82 of 157
Application +4 | -1
If sin⁡α+sin⁡β=a\sin \alpha+\sin \beta=a and cos⁡α−cos⁡β=b\cos \alpha-\cos \beta=b, then tan⁡α−β2\tan \frac{\alpha-\beta}{2} is equal to
View answer and explanation
Correct option: B
−b/a-b / a
Option B (−b/a-b / a) is correct because by using sum-to-product trigonometric identities and dividing the resulting equations, we can find that tan⁡α−β2=−ba\tan \frac{{\alpha-\beta}}{{2}} = -\frac{{b}}{{a}}.
83 Medium
83 of 157
Application +4 | -1
If 0<x<π2,0 < x < \frac{\pi }{2}, then the minimum value of cos⁡3xsin⁡x+sin⁡3xcos⁡x\frac{{{{\cos }^3}x}}{{\sin x}} + \frac{{{{\sin }^3}x}}{{\cos x}} is
View answer and explanation
Correct option: D
1
Option D (1) is correct because the minimum value is found using AM ≥ GM inequality, simplifying to sin⁡2x\sin 2x, and then recognizing that the minimum value of sin⁡2x\sin 2x in the given interval is 1 when x=π/4x = \pi/4.
84 Medium
84 of 157
Application +4 | -1
If tan⁡α=(1+2−x)−1,tan⁡β=(1+2x+1)−1\tan \alpha=\left(1+2^{-x}\right)^{-1}, \tan \beta=\left(1+2^{x+1}\right)^{-1}, then α+β\alpha+\beta equals
View answer and explanation
Correct option: B
π/4\pi / 4
Option B (π4\frac{{\pi}}{{4}}) is correct because by substituting t=2xt = 2^x and using the tangent addition formula, tan⁡(α+β)\tan(\alpha + \beta) simplifies to 1, which means α+β=π4\alpha + \beta = \frac{{\pi}}{{4}}.
85 Medium
85 of 157
Analyzing +4 | -1
If in a △ABC,2cos⁡Aa+cos⁡Bb+2cos⁡Cc=abc+bca\triangle A B C, \frac{2 \cos A}{a}+\frac{\cos B}{b}+\frac{2 \cos C}{c}=\frac{a}{b c}+\frac{b}{c a}, then ∠A\angle A is equal to
View answer and explanation
Correct option: A
90∘90^{\circ}
Option A (90∘90^{\circ}) is correct because simplifying the given equation using the sine rule and cosine rule leads to the conclusion that angle A is 90∘90^{\circ}.
86 Medium
86 of 157
Application +4 | -1
If cos⁡A3=cos⁡B4=15,−π2<(A,B)<0\frac{\cos A}{3}=\frac{\cos B}{4}=\frac{1}{5},-\frac{\pi}{2} < (A, B) < 0, then value of 2sin⁡A+4sin⁡B2 \sin A+4 \sin B is
View answer and explanation
Correct option: C
-4
Option C (-4) is correct because given the conditions, we find sin⁡A=−45\sin A = -\frac{4}{5} and sin⁡B=−35\sin B = -\frac{3}{5}. Substituting these values into 2sin⁡A+4sin⁡B2 \sin A + 4 \sin B gives 2(−45)+4(−35)=−85−125=−42(-\frac{4}{5}) + 4(-\frac{3}{5}) = -\frac{8}{5} - \frac{12}{5} = -4.
87 Medium
87 of 157
Understanding +4 | -1
The value of cos⁡10∘−sin⁡10∘\cos 10^\circ - \sin 10^\circ is
View answer and explanation
Correct option: A
Positive
Option A (Positive) is correct because for angles between 0 and 45 degrees, the cosine function is greater than the sine function.
88 Medium
88 of 157
Application +4 | -1
The value of cos⁡(π/13)+cos⁡(2π/13)+….+cos⁡(12π/13)\cos (\pi / 13)+\cos (2 \pi / 13)+\ldots .+\cos (12 \pi / 13) is equal to
View answer and explanation
Correct option: A
0
Option A (0) is correct because the terms in the cosine series cancel each other out due to the symmetry around π\pi.
89 Medium
89 of 157
Analyzing +4 | -1
If a and b are positive numbers such that a>b,a > b, then the minimum value of a  sec  θ−btan⁡θ(0<θ<π2)\mathrm{a\;sec}\;\theta - b\tan \theta \left( {0 < \theta < \frac{\pi }{2}} \right) is
View answer and explanation
Correct option: D
a2−b2\sqrt {{a^2} - {b^2}}
Option D (a2−b2\sqrt{{{{a^2} - {b^2}}}}) is correct because it represents the minimum value of the expression, found using differentiation and trigonometric identities.
90 Medium
90 of 157
Understanding +4 | -1
Which of the following relations is possible?
View answer and explanation
Correct option: B
  tan  θ=1002\mathrm{\;tan}\,\,\mathrm{\theta } = {100^2}
Option B (  tan  θ=1002\;tan\,\,\theta = {100^2}) is correct because the range of the tangent function is all real numbers.
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