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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Medium

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91 Medium
91 of 157
Analyzing +4 | -1
3 (sin⁡  x−cos⁡  x)4+6  (sin⁡  x+cos⁡  x)2+4    (sin⁡6  x+cos⁡6  x)3\,{(\sin \,\,x - \cos \,\,x)^4} + 6\,\,{\left( {\sin \,\,x + \cos \,\,x} \right)^2} + 4\,\,\;\left( {{{\sin }^6}\,\,x + {{\cos }^6}\,\,x} \right) is equal to
View answer and explanation
Correct option: C
13\mathrm{13}
Option C (13) is correct because after simplification using trigonometric identities, the expression evaluates to 13.
92 Medium
92 of 157
Application +4 | -1
tan⁡100∘+tan⁡125∘+tan⁡100∘tan⁡125∘=\tan 100^{\circ}+\tan 125^{\circ}+\tan 100^{\circ} \tan 125^{\circ}=
View answer and explanation
Correct option: D
1
Option D (1) is correct because using the tangent addition formula, the expression simplifies to 1.
93 Medium
93 of 157
Analyzing +4 | -1
If mm and MM are the minimum and the maximum values of 4+12sin⁡22x−2cos⁡4x,x∈R4+\frac{1}{2} \sin ^{2} 2 x-2 \cos ^{4} x, x \in R, then M−mM-m is equal to:
View answer and explanation
Correct option: B
154\frac{15}{4}
Option B (15/4) is correct because by completing the square and finding the maximum and minimum values of the given expression, the difference M-m equals 15/4.
94 Medium
94 of 157
Application +4 | -1
Which of the following is rational?
View answer and explanation
Correct option: C
sin⁡15∘cos⁡15∘\sin 15^{\circ} \cos 15^{\circ}
Option C (sin⁡15∘cos⁡15∘\sin 15^{\circ} \cos 15^{\circ}) is correct because sin⁡15∘cos⁡15∘=1/4\sin 15^{\circ} \cos 15^{\circ} = 1/4, which is a rational number.
95 Medium
95 of 157
Application +4 | -1
If in a △ABC,(a+b+c)(b+c−a)=λbc\triangle A B C,(a+b+c)(b+c-a)=\lambda b c, then
View answer and explanation
Correct option: D
0<λ<40<\lambda<4
Option D (0<λ<40<\lambda<4) is correct because the given equation can be transformed to relate λ\lambda to cos⁡A\cos A, and the range of cos⁡A\cos A restricts the possible values of λ\lambda.
96 Medium
96 of 157
Analyzing +4 | -1
Let θ∈(0,   π4)\theta \in \left( {0,\;\,\frac{\pi }{4}} \right) and t1=(tan⁡  θ)tan⁡θ,   t2=(tan⁡  θ)cot⁡θ,   t3=(cot⁡  θ)tan⁡θ{t_1} = {\left( {\tan \,\,\theta } \right)^{\tan \theta }},\,\;{t_2} = {\left( {\tan \,\,\theta } \right)^{\cot \theta }},\;\,{t_3} = {\left( {\cot \,\,\theta } \right)^{\tan \theta }} and t4=(cot⁡  θ)tan⁡θ{t_4} = {\left( {\cot \,\,\theta } \right)^{\tan \theta }} , then
View answer and explanation
Correct option: B
t4>t3>t1>t2{t_4} > {t_3} > {t_1} > {t_2}
Option B (t4>t3>t1>t2{t_4} > {t_3} > {t_1} > {t_2}) is correct because when θ∈(0,π4)\theta \in (0, \frac{\pi}{4}), tan⁡θ<1\tan \theta < 1 and cot⁡θ>1\cot \theta > 1, leading to the given inequality.
97 Medium
97 of 157
Application +4 | -1
The value of (1+cos⁡π6)(1+cos⁡π3)(1+cos⁡2π3)\left(1+\cos \frac{\pi}{6}\right)\left(1+\cos \frac{\pi}{3}\right)\left(1+\cos \frac{2 \pi}{3}\right) (1+cos⁡7π6)\left(1+\cos \frac{7 \pi}{6}\right) is
View answer and explanation
Correct option: A
3/163 / 16
Option A (3/163 / 16) is correct because substituting the values of the cosine functions and simplifying the expression leads to the result 3/163/16.
98 Medium
98 of 157
Analyzing +4 | -1
If sec⁡  α\sec \,\,\alpha and cosec    α\mathrm{cosec}\;\,\,\alpha are the roots of the equation x2−ax+b=0{x^2} - ax + b = 0, then
View answer and explanation
Correct option: B
a2=b(b+2){a^2} = b\left( {b + 2} \right)
Option B (a2=b(b+2){a^2} = b(b + 2)) is correct because by using Vieta's formulas to find the sum and product of the roots and then applying the trigonometric identity sin⁡2α+cos⁡2α=1{\sin^2 \alpha + \cos^2 \alpha = 1}, we can derive the given relation.
99 Medium
99 of 157
Application +4 | -1
The value of cos⁡4(π8)+cos⁡4(3π8)+cos⁡4(5π8)+cos⁡4\cos ^4\left(\frac{\pi}{8}\right)+\cos ^4\left(\frac{3 \pi}{8}\right)+\cos ^4\left(\frac{5 \pi}{8}\right)+\cos ^4 (7π8)\left(\frac{7 \pi}{8}\right) is
View answer and explanation
Correct option: C
3/23 / 2
Option C (3/2) is correct because by using trigonometric identities and simplifying, the expression cos⁡4(π8)+cos⁡4(3π8)+cos⁡4(5π8)+cos⁡4(7π8)\cos^4(\frac{\pi}{8}) + \cos^4(\frac{3\pi}{8}) + \cos^4(\frac{5\pi}{8}) + \cos^4(\frac{7\pi}{8}) simplifies to 32\frac{3}{2}.
100 Medium
100 of 157
Analyzing +4 | -1
If   2sin⁡  α1+cos⁡  α+sin⁡  α=x,    then   1−cos⁡  α−sin⁡  αcos⁡  α  \;\frac{{2\sin \,\,\alpha }}{{1 + \cos \,\,\alpha + \sin \,\,\alpha }} = x,\,\,\;\mathrm{then}\;\,\frac{{1 - \cos \,\,\alpha - \sin \,\,\alpha }}{{\cos \,\,\alpha }}\; is equal to

View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the given expression simplifies to −x-x, which is not among the provided options.
101 Medium
101 of 157
Application +4 | -1
If the line of action of the resultant of two forces PP and QQ divides the angle between them in the ratio 1: 2 then the magnitude of resultant is
View answer and explanation
Correct option: D
P2−Q2Q\frac{{P}^2-{Q}^2}{{Q}}
Option D (P2−Q2Q\frac{{P}^2-{Q}^2}{{Q}}) is correct because it is derived by using the sine rule and trigonometric identities to relate the forces and the angles.
102 Medium
102 of 157
Application +4 | -1
Range of cos⁡2x−10cos⁡x+25+2\sqrt{\cos ^2 x-10 \cos x+25}+2 is
View answer and explanation
Correct option: C
[6,8][6,8]
Option C ([6,8]) is correct because simplifying the expression cos⁡2x−10cos⁡x+25+2\sqrt{{\cos ^2 x-10 \cos x+25}}+2 leads to the range [6,8].
103 Medium
103 of 157
Application +4 | -1
If 0<x<π0 < x < \pi and cos⁡x+sin⁡x=12\cos x+\sin x=\frac{1}{2}, then tan⁡x\tan \mathrm{x} is
View answer and explanation
Correct option: C
−(4+7)3-\frac{(4+\sqrt{7})}{3}
Option C is correct because solving the trigonometric equation and considering the domain 0 < x < π leads to the correct value for tan x.
104 Medium
104 of 157
Application +4 | -1
If cos⁡xa=sin⁡xb\frac{\cos x}{a}=\frac{\sin x}{b} then ∣acos⁡2x+bsin⁡2x∣|a \cos 2 x+b \sin 2 x| is equal to
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the expression simplifies to ∣a∣|a| using trigonometric identities and the given condition.
105 Medium
105 of 157
Application +4 | -1
2sin⁡(π22)sin⁡(3π22)sin⁡(5π22)sin⁡(7π22)sin⁡(9π22)2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right) is equal to:
View answer and explanation
Correct option: B
116\frac{1}{16}
Option B (116\frac{1}{16}) is correct because by using trigonometric identities, the expression simplifies to 116\frac{1}{16}.

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106 Medium
106 of 157
Application +4 | -1
What is cos⁡20∘+cos⁡100∘+cos⁡140∘\cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ} equal to ?
View answer and explanation
Correct option: D
0
Option D (0) is correct because using the sum-to-product trigonometric identity, the expression simplifies to zero.
107 Medium
107 of 157
Application +4 | -1
If 12cot⁡2θ−31cosec⁡θ+32=012 \cot ^2 \theta-31 \operatorname{cosec} \theta+32=0, then the value of sin⁡θ\sin \theta is
View answer and explanation
Correct option: C
4/54 / 5 or 3/43 / 4
Option C (4/54/5 or 3/43/4) is correct because solving the trigonometric equation 12cot⁡2θ−31cosec⁡θ+32=012 \cot ^2 \theta-31 \operatorname{cosec} \theta+32=0 leads to sin⁡θ=3/4\sin \theta = 3/4 or 4/54/5.
108 Medium
108 of 157
Application +4 | -1
If xcos⁡θ=ycos⁡(θ+2π3)=zcos⁡(θ−2π3)\frac{x}{\cos \theta}=\frac{y}{\cos \left(\theta+\frac{2 \pi}{3}\right)}=\frac{z}{\cos \left(\theta-\frac{2 \pi}{3}\right)}, then x+y+zx+y+z is equal to
View answer and explanation
Correct option: C
0
Option C (0) is correct because using the given condition and the cosine addition formula, the expression simplifies to zero.
109 Medium
109 of 157
Analyzing +4 | -1
If sin⁡A+sin⁡B+sin⁡C=3\sin A+\sin B+\sin C=3, then cos⁡A+cos⁡B+cos⁡C\cos A+\cos B+\cos C is equal to
View answer and explanation
Correct option: D
0
Option D is correct because if the sum of sines is 3, each sine must be 1, making each angle π2\frac{\pi}{2}, thus each cosine is 0.
110 Medium
110 of 157
Application +4 | -1
In a △ABC\triangle A B C, if a,ba, b and cc are in AP, then the value of sin⁡A2sin⁡C2sin⁡B2\frac{\sin \frac{A}{2} \sin \frac{C}{2}}{\sin \frac{B}{2}} is
View answer and explanation
Correct option: B
12\frac{1}{2}
Option B (12\frac{1}{2}) is correct because when a,b,ca, b, c are in A.P., we can express ss in terms of bb, and then simplify the given expression using the properties of sine and the semi-perimeter.
111 Medium
111 of 157
Application +4 | -1
If sin⁡x+sin⁡2x+sin⁡3x=1\sin x+\sin ^2 x+\sin ^3 x=1 then evaluate the value of cos⁡6x−4cos⁡4x+8cos⁡2x\cos ^6 x-4 \cos ^4 x+8 \cos ^2 x
View answer and explanation
Correct option: A
4
Option A (4) is correct because by substituting sin⁡x+sin⁡2x+sin⁡3x=1\sin x + \sin^2 x + \sin^3 x = 1 and using trigonometric identities, the expression cos⁡6x−4cos⁡4x+8cos⁡2x\cos^6 x - 4 \cos^4 x + 8 \cos^2 x simplifies to 4.
112 Medium
112 of 157
Application +4 | -1
If x,y,zx, y, z are in A.P. then sin⁡x−sin⁡zcos⁡z−cos⁡x\frac{\sin x-\sin z}{\cos z-\cos x} is equal to -
View answer and explanation
Correct option: B
cot⁡y\cot y
Option B (cot⁡y\cot y) is correct because using the A.P. property and trigonometric identities simplifies the expression to cot⁡y\cot y.
113 Medium
113 of 157
Application +4 | -1
cos⁡12∘−sin⁡12∘cos⁡12∘+sin⁡12∘+sin⁡147∘cos⁡147∘=\frac{\cos 12^{\circ}-\sin 12^{\circ}}{\cos 12^{\circ}+\sin 12^{\circ}}+\frac{\sin 147^{\circ}}{\cos 147^{\circ}}=
View answer and explanation
Correct option: C
0
Option C (0) is correct because after simplifying using trigonometric identities, the expression equals 0.
114 Medium
114 of 157
Analyzing +4 | -1
In a △ABC,tan⁡A⋅tan⁡B⋅tan⁡C=9\triangle A B C, \tan A \cdot \tan B \cdot \tan C=9. For such triangles, if tan⁡2A+tan⁡2B+tan⁡2C=k\tan ^2 A+\tan ^2 B+\tan ^2 C=k, then
View answer and explanation
Correct option: B
k>27k>27
Option B (k>27k>27) is correct because using the AM-GM inequality and the fact that tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A + \tan B + \tan C = \tan A \tan B \tan C in a triangle, we can show that tan⁡2A+tan⁡2B+tan⁡2C>27\tan^2 A + \tan^2 B + \tan^2 C > 27.
115 Medium
115 of 157
Application +4 | -1
If x=h+asec⁡θx=h+a \sec \theta and y=k+bcosec⁡θy=k+b \operatorname{cosec} \theta. Then,
View answer and explanation
Correct option: B
a2(x−h)2+b2(y−k)2=1\frac{a^2}{(x-h)^2}+\frac{b^2}{(y-k)^2}=1
Option A (0) is correct because the given conditions imply that the determinant is a skew-symmetric matrix of order 3, which is always 0.
116 Medium
116 of 157
Application +4 | -1
If in a △ABC,sin⁡2A+sin⁡2B+sin⁡2C=2\triangle A B C, \sin ^2 A+\sin ^2 B+\sin ^2 C=2, then the triangle is always
View answer and explanation
Correct option: B
right angled
Option B (right angled) is correct because the given condition sin⁡2A+sin⁡2B+sin⁡2C=2\sin^2 A + \sin^2 B + \sin^2 C = 2 implies that one of the angles must be 90∘90^\circ, making the triangle right-angled.
117 Medium
117 of 157
Application +4 | -1
If a =π/18=\pi / 18 rad, then cos⁡a+cos⁡2a+….+cos⁡18a\cos a+\cos 2 a+\ldots .+\cos 18 a is equal to
View answer and explanation
Correct option: A
0
Option A (0) is correct because the sum telescopes to 0 due to the symmetry around π2\frac{\pi}{2}.
118 Medium
118 of 157
Application +4 | -1
The equation a  cos⁡  θ+b  sin⁡  θ=ca\,\,\cos \,\,\theta + b\,\,\sin \,\,\theta = c has a solution, when a,   ba,\;\,b and cc are real numbers such that

View answer and explanation
Correct option: C
c2≤a2+b2{c^2} \le {a^2} + {b^2}
Option C (c2≤a2+b2{c^2} \le {a^2} + {b^2}) is correct because the equation acos⁡θ+bsin⁡θ=ca\cos \theta + b\sin \theta = c has a solution only if cc lies within the range [−a2+b2,a2+b2][-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}], which implies c2≤a2+b2{c^2} \le {a^2} + {b^2}.
119 Medium
119 of 157
Application +4 | -1
Value of 2(sin⁡6θ+cos⁡6θ)2\left(\sin ^{6} \theta+\cos ^{6} \theta\right) −3(sin⁡4θ+cos⁡4θ)+1-3\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 and simplifying, the expression evaluates to 0.
120 Medium
120 of 157
Analyzing +4 | -1
If θ=2sin⁡x1+sin⁡x+cos⁡x\theta = \frac{{2\sin x}}{{1 + \sin x + \cos x}}, then 1+sin⁡x−cos⁡x1+sin⁡x\frac{{1 + \sin x - \cos x}}{{1 + \sin x}} equals
View answer and explanation
Correct option: C
θ\theta
Option C (θ\theta) is correct because by using trigonometric identities, the expression can be simplified to be equal to the given value of θ\theta.
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