Trigonometric Angles and Equations

On simplifying $${{{{\sin }^3}A + \sin 3A} \over {\sin A}} + {{{{\cos }^3}A - \cos 3A} \over {\cos A}}$$ we get

If $$\sin x + \sin y = a$$ and $$\cos x + \cos y = b$$, then $${\tan ^2}\left( {{{x + y} \over 2}} \right) + {\tan ^2}\left( {{{x - y} \over 2}} \right)$$ is equal to

If $$p = \tan \left( { - {{11\pi } \over 6}} \right)$$, $$q = \tan \left( {{{21\pi } \over 4}} \right)$$ and $$r = \cot \left( {{{283\pi } \over 6}} \right)$$, then which of the following is/are correct?

1. The value of p $$\times$$ r is 2.

2. p, q and r are in GP.

Select the correct answer using the code given below.

$$\sin A + 2\sin 2A + \sin 3A$$ is equal to which of the following?

I. $$4\sin 2A{\cos ^2}\left( {{A \over 2}} \right)$$

II. $$2\sin 2A{\left( {\sin {A \over 2} + \cos {A \over 2}} \right)^2}$$

III. $$8\sin A\cos A{\cos ^2}\left( {{A \over 2}} \right)$$

Select the correct answer using the code given below.

If $$x = \sin 70^\circ \sin 50^\circ $$ and $$y = \cos 60^\circ \cos 80^\circ $$, then what is xy equal to?

If $$\sin {\theta _1} + \sin {\theta _2} + \sin {\theta _3} + \sin {\theta _4} = 4$$, then what is the value of $$\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3} + \cos {\theta _4}$$ ?

What is the value of $$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)$$ ?

If $$x\cos \theta + y\sin \theta = z$$, then what is the value of $${(x\sin \theta - y\cos \theta )^2}$$ ?

If $$\sin 18^\circ = {{\sqrt 5 - 1} \over 4}$$, then what is the value of sin81$$^\circ$$ ?

What is $${{1 - \tan 2^\circ \cot 62^\circ } \over {\tan 152^\circ - \cot 88^\circ }}$$ equal to?

What is the maximum value of the function $$f(\theta )$$ ?

What is the minimum value of the function f($$\theta$$) ?

Consider the following statements

1. $$f(\theta ) = 2$$ has no solution.

2. $$f(\theta ) = {7 \over 2}$$ has a solution.

Which of the above statement(s) is/are correct?

What is the value of $$\sin {A \over 2}\sin {B \over 2}\sin {C \over 2}$$ ?

What is the value of $$\cos \left( {{{A + B} \over 2}} \right)\cos \left( {{{B + C} \over 2}} \right)\cos \left( {{{C + A} \over 2}} \right)$$ ?

What is tan ($$\alpha$$ + $$\beta$$) equal to?

What is $$\sin (\alpha + \beta )sec\alpha sec\beta $$ equal to?

If the equation possesses solution, then what is the minimum value of k?

If the equation possesses solution, then what is the maximum value of k?

Consider the following statements

1. If ABC is an equilateral triangle, then $$3\tan (A + B)\tan C = 1$$.

2. If ABC is a triangle in which A = 78$$^\circ$$, B = 66$$^\circ$$, then $$\tan \left( {{A \over 2} + C} \right) < \tan A$$

3. If ABC is any triangle, then $$\tan \left( {{{A + B} \over 2}} \right)\sin \left( {{C \over 2}} \right) < \cos \left( {{C \over 2}} \right)$$

Which of the above statements is/are correct?

If $$A = (\cos 12^\circ - \cos 36^\circ )(\sin 96^\circ + \sin 24^\circ )$$ and $$B = (\sin 60^\circ - \sin 12^\circ )(\cos 48^\circ - \cos 72^\circ )$$, then what is $${A \over B}$$ equal to?

The value of $$\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ $$ is equal to

The value of $$\sqrt 3 \cos ec20^\circ - \sec 20^\circ $$ is equal to

Angle $$\alpha$$ is divided into two parts A and B such that A $$-$$ B = x and tan A : tan B = p : q. The value of sin x is equal to

$$\sqrt {1 + \sin A} = - \left( {\sin {A \over 2} + \cos {A \over 2}} \right)$$ is true, if

In triangle ABC, if $${{{{\sin }^2}A + {{\sin }^2}B + {{\sin }^2}C} \over {{{\cos }^2}A + {{\cos }^2}B + {{\cos }^2}C}} = 2$$, then the triangle is

Match List-I with List-II and select the correct answer using the code given below the lists

Image

If $$\sin A = {3 \over 5}$$, where $$450^\circ < A < 540^\circ $$, then $$\cos {A \over 2}$$ is equal to

What is $${1 \over {\sin 10^\circ }} - {{\sqrt 3 } \over {\cos 10^\circ }}$$ equal to?

The maximum value of $$\sin \left( {x + {\pi \over 6}} \right) + \cos \left( {x + {\pi \over 6}} \right)$$ in the interval $$\left( {0,{\pi \over 2}} \right)$$ is attained at

If $$K = \sin \left( {{\pi \over {18}}} \right)\sin \left( {{{5\pi } \over {18}}} \right)\sin \left( {{{7\pi } \over {18}}} \right)$$, then what is the value of K?

The expression $${{\sin \alpha + \sin \beta } \over {\cos \alpha + \cos \beta }}$$ is equal to

If $$\sin \theta = 3\sin (\theta + 2\alpha )$$, then the value of tan ($$\theta$$ + $$\alpha$$) + 2 tan$$\alpha$$ is equal to

What is the value of tan18$$^\circ$$ ?

If $$\tan (\alpha + \beta ) = 2$$ and $$\tan (\alpha - \beta ) = 1$$, then tan(2$$\alpha$$) is equal to

Consider the following for triangle ABC

1. $$\sin \left( {{{B + C} \over 2}} \right) = \cos \left( {{A \over 2}} \right)$$

2. $$\tan \left( {{{B + C} \over 2}} \right) = \cot \left( {{A \over 2}} \right)$$

3. $$\sin (B + C) = \cos A$$

4. $$\tan (B + C) = - \cot A$$

Which of the above are correct?

If $$\sec \theta - \cos ec\theta = {4 \over 3}$$, then what is (sin$$\theta$$ $$-$$ cos$$\theta$$) equal to?

What is the maximum value of the function f(x) = 4sin² x + 1 ?

What is $${{2\tan \theta } \over {1 + {{\tan }^2}\theta }}$$ equal to?

If $$\sec (\theta - \alpha )$$, $$\sec \theta $$ and $$\sec (\theta + \alpha )$$ are in AP, where cos $$\alpha$$ $$\ne$$ 1, then what is the value of sin²$$\theta$$ + cos$$\alpha$$ ?

If A + B + C = 180$$^\circ$$, then what is $$\sin 2A - \sin 2B - \sin 2C$$ equal to?

A is an angle in the fourth quadrant it satisfies the trigonometric equation $$3{(3 - {\tan ^2}A - \cot A)^2} = 1$$. Which one of the following is a value of A?

What is/are the solution(s) of the trigonometric equation $$\cos ec\,x + \cos x = \sqrt 3 $$ where 0 < x < 2$$\pi$$ ?

If $$\theta = {\pi \over 8}$$, then what is the value of $${(2\cos \theta + 1)^{10}}{(2\cos 2\theta - 1)^{10}}{(2\cos \theta - 1)^{10}}{(2\cos 4\theta - 1)^{10}}$$ ?

If cos $$\alpha$$ and cos $$\beta$$ (0 < $$\alpha$$ < $$\beta$$ < $$\pi$$) are the roots of the quadratic equation 4x² $$-$$ 3 = 0, then what is the value of sec $$\alpha$$ $$\times$$ sec $$\beta$$ ?

If $$A = {\sin ^2}\theta + {\cos ^4}\theta $$, then for all real $$\theta$$, which one of the following is correct?

What is $${{\sin 5x - \sin 3x} \over {\cos 5x + \cos 3x}}$$ equal to?

What is $$\sin 105^\circ + \cos 105^\circ $$ equal to?

If x, x $$-$$ y and x + y are the angles of a triangle (not an equilateral triangle) such that tan (x $$-$$ y), tan x and tan (x + y) are in GP, then what is x equal to?

ABC is a triangle inscribed in a circle with centre O. Let $$\alpha$$ = $$\angle$$BAC, where 45$$^\circ$$ < $$\alpha$$ < 90$$^\circ$$. Let $$\beta$$ = $$\angle$$BOC. Which one of the following is correct?

If $${{\sin (x + y)} \over {\sin (x - y)}} = {{a + b} \over {a - b}}$$, then what is $${{\tan x} \over {\tan y}}$$ equal to?

If sin$$\alpha$$ + sin$$\beta$$ = 0 = cos$$\alpha$$ + cos$$\beta$$, where 0 < $$\beta$$ < $$\alpha$$ < 2$$\pi$$, then which one of the following is correct?

Suppose cos A is given. If only one value of $$\cos \left( {{A \over 2}} \right)$$ is possible, then A must be

If $$\cos \alpha + \cos \beta + \cos \gamma = 0$$, where $$0 < \alpha \le {\pi \over 2}$$, $$0 < \beta \le {\pi \over 2}$$, $$0 < \gamma \le {\pi \over 2}$$, then what is the value of sin$$\alpha$$ + sin$$\beta$$ + sin$$\gamma$$ ?

The maximum value of $$\sin \left( {x + {\pi \over 5}} \right) + \cos \left( {x + {\pi \over 5}} \right)$$, where $$x \in \left( {0,{\pi \over 2}} \right)$$, is attained at

What is the maximum value of $$16\sin \theta - 12{\sin ^2}\theta $$ ?

What is tan25$$^\circ$$ tan15$$^\circ$$ + tan15$$^\circ$$ tan50$$^\circ$$ + tan25$$^\circ$$ tan50$$^\circ$$ equal to?

What is $$\cot A + \cos ecA$$ equal to?

What is $$\cot \left( {{A \over 2}} \right)$$ $$-$$ $$\tan \left( {{A \over 2}} \right)$$ equal to?

If angle C of a triangle ABC is a right angle, then what is tan A + tan B equal to?

What is $$\cos 80^\circ + \cos 40^\circ - \cos 20^\circ $$ equal to?

If $$2\tan A = 3\tan B = 1$$, then what is $$\tan (A - B)$$ equal to?

If $$\tan A - \tan B = x$$ and $$\cot B - \cot A = y$$, then what is the value of $$\cot (A - B)$$ ?

What is the minimum value of $${{{a^2}} \over {{{\cos }^2}x}} + {{{b^2}} \over {{{\sin }^2}x}}$$ where a > 0 and b > 0 ?

What is $$\sin (\alpha + \beta ) - 2sin\alpha cos\beta + sin(\alpha - \beta )$$ equal to?

If $$\cos ec\theta = {{29} \over {21}}$$, where 0 < $$\theta$$ < 90$$^\circ$$, then what is the value of 4 sec$$\theta$$ + 4 tan$$\theta$$ ?

Consider the following statements

1. cos $$\theta$$ + sec $$\theta$$ can never be equal to 1.5 .

2. tan $$\theta$$ + cot $$\theta$$ can never be less than 2 .

Which of the above statements is/are correct?

What is the length of the chord of a unit circle which subtends an angle $$\theta$$ at the centre?

What is the least value of $$25\cos e{c^2}x + 36{\sec ^2}x$$ ?

What is the value of $${{\sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ } \over {\cos 28^\circ \cos 88^\circ + \cos 178^\circ \sin 208^\circ }}$$ ?

tan54$$^\circ$$ can be expressed as

What is the value of $$\theta$$ ?

What is the value of A?

What is the value of B?

What is $$\cos (\alpha - \beta )$$ equal to?

What is $${\sin ^2}(\alpha - \beta ) + 2ab\cos (\alpha - \beta )$$ equal to?

If $$\sin \alpha + \cos \alpha = p$$, then what is $${\cos ^2}(2\alpha )$$ equal to?

If $$\tan \theta = {1 \over 2}$$ and $$\tan \phi = {1 \over 3}$$, then what is the value of $$(\theta + \phi )$$ ?

If $$\cos A = {3 \over 4}$$, then what is the value of $$\sin \left( {{A \over 2}} \right)\sin \left( {{{3A} \over 2}} \right)$$ ?

What is the value of $$\tan 75^\circ + \cot 75^\circ $$ ?

What is the value of $$\cos 46^\circ \cos 47^\circ \cos 48^\circ \cos 49^\circ \cos 50^\circ .......\cos 135^\circ $$ ?

If $$\sin 2\theta = \cos 3\theta $$, where $$0 < \theta < {\pi \over 2}$$, then what is sin$$\theta$$ equal to?

If the roots of the equation $${x^2} + px + q = 0$$ are tan 19$$^\circ$$ and tan 26$$^\circ$$, then which one of the following is correct?

What is $${(1 + \tan \alpha \tan \beta )^2} + {(\tan \alpha - \tan \beta )^2} - {\sec ^2}\beta {\sec ^2}\beta $$ equal to?

If $$\sin \theta + \cos \theta = \sqrt 2 \cos \theta $$, then what is $$(\cos \theta - \sin \theta )$$ equal to?

If $$\sin \theta = - {1 \over 2}$$ and $$\tan \theta = {1 \over {\sqrt 3 }}$$, then in which quadrant does $$\theta$$ lie?

If $$p = \cos ec\theta - \cot \theta $$ and $$q = {(\cos ec\theta + \cot \theta )^{ - 1}}$$, then which one of the following is correct?

The roots of the equation 7x² - 6x + 1 = 0 are tan α and tan β, where 2α and 2β are the angles of a triangle. Which one of the following is correct?

What is the number of solutions of the equation cot2x cot3x = 1 for 0 < x

What is the general solution of cos¹⁰⁰ x - sin¹⁰⁰ x = 1?

What is sin12° sin48° equal to?

What is $\rm \frac{\cos 17^\circ-\sin 17^\circ}{\cos 17^\circ+\sin ^\circ}$ equal to

Consider the following numbers :

I. tan 22.5°

II. cot 22.5°

III. tan 22.5° - cot 22.5°

How many of the above are irrational numbers?

If $\rm \frac{x}{\cos \theta}=\frac{y}{\cos \left(\frac{2\pi}{3}-\theta\right)}=\frac{z}{\cos\left(\frac{2\pi}{3}+\theta\right)}$ then what is x + y + z equal to?

If p tan (θ - 30°) = q tan (θ + 120°), then what is (p + q) / (p - q) equal to?

What is the maximum value of a cos x + b sin x + c?

$$ \text { What is } \frac{\sqrt{3} \cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ} \cos 25^{\circ}} \text { equal to ? } $$

What is $\sin 9^\circ - \cos 9^\circ$ equal to?

100

What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi < \theta < \pi$?

101

What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot \sin10^{\circ}$ equal to?

102

If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$?

103

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos(2 \alpha+2 \beta)$?

104

What is the value of $\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$?

If tan (π cos θ) = cot (π sin θ), $0<\theta<\frac{\pi}{2}$; then what is the value of $8 \sin ^2\left(\theta+\frac{\pi}{4}\right)$ ?

105

If $\tan \alpha=\frac{1}{7}$, $\sin \beta=\frac{1}{\sqrt{10}}$; $0<\alpha, \beta<\frac{\pi}{2}$, then what is the value of cos (α + 2β) ?

106

What is the period of the function?

107

What is the value of p + q?

108

109

What is the value of pq?

110

What is pq equal to ?

For how many values of x does $\frac{1}{p}$ become zero?

111

What is a value of sin 3x + sin 3y?

112

What is a value of cos³ x + cos³ y?

113

What is cos 2β equal to ?

114

What is the value of sec²γ?

115

What is the minimum value of x ?

116

At what value of A does x attain the minimum value ?

117

The given equation can be reduced to

118

If sin2x = a - b√c, where a and b are natural numbers and c is prime number, then what is the value of a - b + 2c ?

119

What is the least value of m(θ) ?

120

Under what condition does m attain the least value ?

121

If f(x) = x(4x² - 3), then what is f(sinθ) equal to ?

122

What is the value of cosec$\left(−\frac{73 \pi}{3}\right)$ ?

123

What is the value of $\cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)$ ?

124

What is the value of tan$\left(\frac{3\pi}{8}\right)$ ?

125

What is cos 36° − cos 72° equal to ?

126

If sec x = $\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of tan x + sin x ?

127

What is the value of tan2 165° + cot2 165° ?

128

What is the value of $\sin\left(2\text{n}\pi+\frac{5\pi}{6}\right)\sin\left(2\text{n}\pi−\frac{5\pi}{6}\right)$ where n ∈ Z ?

129

If 1 + 2(sin x + cos x)(sin x − cos x) = 0 where 0 < x < 360°, then how many values does x take ?

130

What is the maximum value of 3(sin x − cos x) + 4(cos³ x − sin³ x) ?

131

If f(α) = $\sqrt{\sec^2\alpha−1}$, then what is $\frac{f(\alpha)+f(\beta)}{1−f(\alpha) f(\beta)}$ equal to ?

132

What is value of cot² 15° + tan² 15° ?

133

In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?

134

If $α + β = \frac{\pi}{4}$ and 2tan α = 1, then what is tan 2β equal to?

135

If tan (45° + θ) = 1 + sin 2θ, where $-\frac{\pi}{4}< θ <\frac{\pi}{4},$ then what is the value of cos 2θ?

136

Let sin 2θ = cos 3θ, where θ is acute angle. What is the value of 1 + 4 sin θ? (given that $\sin 18^\circ =\frac{\sqrt 5 - 1}{4}$)

137

If $\tan θ = - \frac{5}{12},$ then what can be the value of sin θ?

138

What is the value of $\cos^4\frac{7\pi}{8}+\cos^4\frac{5\pi}{8}?$

139

What is $\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)$ equal to?

140

If tan α + tan β = 1 - tan α.tan β, where tan α tan β ≠ 1, then which of the following is one of the values of (α + β) ?

141

If (1 + tan θ) (1 + tan 9θ) = 2, then what is the value of tan(10θ)?

142

What is the value of sin 0° + sin 10° + sin 20° + sin 30° + ⋯ + sin 360°?

143

What is the maximum value of the functions $f(x) = \frac{1}{\tan x+\cot x},$ where $0 < x < \frac{\pi}{2}?$

144

If sinθ and cosθ are the roots of the equation ax² + bx + c = 0, then which one of the following is correct?

145

If sin(A + B) = 1 and 2 sin(A - B) = 1, where 0 < A, B < $\frac{\pi}{2}$, then what is tan A ∶ tan B equal to?

146

Let sin x + sin y = cos x + cos y for all x, y ∈ ℝ. What is $\rm tan \left(\frac{x}{2}+\frac{y}{2}\right)$ equal to?

147

Let 4sin² x = 3, where 0 ≤ x ≤ π. What is tan3x equal to?

148

Let the equation sec x.cosec x = p have a solution, where p is a positive real number. What should be the smallest value of p?

149

For what value of θ, where 0 < θ < $\frac{\pi}{2}$, does sin θ + sin θ cos θ maximum value?

150

Let ABC be a triangle. If cos2A + cos2B + cos2C = -1 then which one of the following is correct?

151

If $\tan A=\dfrac{1}{7}$, then what is cos 2A equal to?

152

If $\rm tan\:x=-\dfrac{3}{4}$ and x is in the second quadrant, then what is the value of sin x ⋅ cos x?

153

154

What is the value of the following?

$\rm cosec\left(\dfrac{7\pi}{6}\right)sec\left(\dfrac{5\pi}{3}\right)$

155

What is the value of the following?

tan 31° tan 33° tan 35° _ _ _ _ _ tan 57° tan 59°

156

What is the value of the following?

(sin 24° + cos 66°)(sin 24° - cos 66°)

157

What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?

158

What is $\rm \frac{1+tan^2\theta}{1+cot^2\theta}-\left(\frac{1-tan\theta}{1-cot\theta}\right)^2$equal to?

If 7 sinθ + 24 cosθ = 25, then what is the value of (sin θ + cos θ)?

159

If 3cosθ = 4sinθ, then what is the value of tan (45° + θ)?

160

The number of integer values of k, for which the equation 2sinx = 2k + 1 has a solution, is

161

162

What is cot 2x cot 4x - cot 4x cot 6x - cot 6x cot 2x equal to