Practice MCQ Questions and Answer on Trigonometry

341.

If θ is a acute angle and sin(θ + 18°) = $$\frac{1}{2}\text{,}$$ then the value of θ in circular measure is?

• (A) $$\frac{\pi }{12}$$ Radians
• (B) $$\frac{\pi }{15}$$ Radians
• (C) $$\frac{2\pi }{5}$$ Radians
• (D) $$\frac{3\pi }{13}$$ Radians

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 $$\eqalign{ & {\text{sin}}\left( {\theta + {{18}^ \circ }} \right){\text{ = }}\frac{1}{2} \cr & {\text{sin}}\left( {\theta + {{18}^ \circ }} \right) = {\text{sin }}{30^ \circ } \cr & \theta + {18^ \circ } = {30^ \circ } \cr & \therefore \theta = {12^ \circ } \cr & {\text{We know that,}} \cr & {180^ \circ } = \pi \cr & {12^ \circ } = \frac{\pi }{{{{180}^ \circ }}} \times 12 = \frac{\pi }{{15}} \cr} $$

342.

If α + β = 90° and α : β = 2 : 1, then the ratio of cosα to cosβ is?

• (A) 1 : $$\sqrt{3}$$
• (B) 1 : 3
• (C) 1 : $$\sqrt{2}$$
• (D) 1 : 2

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 $$\eqalign{ & \alpha + \beta = {90^ \circ } \cr & {\text{and }}\alpha :\beta = 2:1 \cr & 2x + x = {90^ \circ } \cr & x = {30^ \circ } \cr & {\text{ }}\alpha = {60^ \circ } \cr & \beta = {30^ \circ } \cr & \frac{{{\text{cos }}\alpha }}{{\cos \beta }} \cr & = \frac{{\cos {{60}^ \circ }}}{{\cos {{30}^ \circ }}} \cr & = \frac{{\frac{1}{2}}}{{\frac{{\sqrt 3 }}{2}}} \cr & = \frac{1}{2} \times \frac{2}{{\sqrt 3 }} \cr & = \frac{1}{{\sqrt 3 }} \cr & = 1:\sqrt 3 \cr} $$

343.

What will be the value of sin10° - $$\frac{4}{3}$$sin³10°?

• (A) $$\frac{1}{3\sqrt{3}}$$
• (B) $$\frac{1}{6}$$
• (C) $$\frac{1}{2\sqrt{3}}$$
• (D) $$\frac{\sqrt{3}}{6}$$

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 $$\eqalign{ & \sin 10 - \frac{4}{3}{\sin ^3}10 \cr & = \frac{{3\sin 10 - 4{{\sin }^3}10}}{3} \cr & = \frac{1}{3}\sin 3 \times {10^ \circ }\,\,\,\,\,\left[ {\because \sin A = 3\sin A - 4{{\sin }^3}A} \right] \cr & = \frac{1}{3} \times \frac{1}{2} \cr & = \frac{1}{6} \cr} $$

344.

What is the value of cosec(65° + θ) - sec(25° - θ) + tan²20° - cosec²70°?

• (A) -1
• (B) 0
• (C) 1
• (D) 2

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 cosec(65° + θ) - sec(25° - θ) + tan220° - cosec270° = sec(25° - θ) - sec(25° - θ) + tan220° - sec220° = tan220° - sec220° = -1

345.

The value of, $$\text{sec}\theta \left(\frac{1+\sin \theta }{\text{cos}\theta }+\frac{\text{cos}\theta }{1+\sin \theta }\right)$$      - $$2\text{ta}{\text{n}}^2\theta$$   is?

• (A) 4
• (B) 1
• (C) 2
• (D) 0

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 $$\eqalign{ & {\text{sec}}\theta \left( {\frac{{1 + \sin \theta }}{{{\text{cos}}\theta }} + \frac{{{\text{cos}}\theta }}{{1 + \sin \theta }}} \right) - 2{\text{ta}}{{\text{n}}^2}\theta \cr & {\bf{Shortcut method:}} \cr & {\text{Take, }}\theta = {0^ \circ } \cr & \Rightarrow {\text{sec }}{0^ \circ }\left( {\frac{{1 + \sin {0^ \circ }}}{{{\text{cos }}{0^ \circ }}} + \frac{{{\text{cos }}{0^ \circ }}}{{1 + \sin {0^ \circ }}}} \right) - 2{\text{ta}}{{\text{n}}^2}{0^ \circ } \cr & \Rightarrow 1\left( {\frac{{1 + 0}}{1} + \frac{1}{{1 + 0}}} \right) - 0 \cr & \Rightarrow 2 \cr} $$

346.

If sinθ × cosθ = $$\frac{1}{2}\text{,}$$ Then the value of sinθ - cosθ is where 0° θ 90°.

• (A) 0
• (B) $$\sqrt{2}$$
• (C) 2
• (D) 1

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 $$\eqalign{ & \sin \theta \times \cos \theta = \frac{1}{2} \cr & {\text{Multiply by 2 both side}} \cr & 2\sin \theta \times \cos \theta = 1 \cr & \sin 2\theta = 1 \cr & \sin 2\theta = \sin {90^ \circ } \cr & 2\theta = {90^ \circ } \cr & \theta = {45^ \circ } \cr & {\text{So,}}\sin \theta - {\text{cos}}\theta \cr & = \sin {45^ \circ } - \cos {45^ \circ } \cr & = \frac{1}{{\sqrt 2 }} - \frac{1}{{\sqrt 2 }} \cr & = 0{\text{ }} \cr} $$

347.

The value of following is, cos24° + cos55° + cos125° + cos204° + cos300° ?

• (A) $$-\frac{1}{2}$$
• (B) $$\frac{1}{2}$$
• (C) 2
• (D) 1

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 The value of, cos24° + cos55° + cos125° + cos204° + cos300° We know that, cos(180° $$ \pm $$ θ) = -cosθ ⇒ cos24° + cos55° + cos (180° - 55°) + cos (180° + 24°) + cos (360° - 60°) ⇒ cos24° + cos55° - cos55° - cos24° + cos60° ⇒ cos60° ⇒ $$\frac{1}{2}$$

348.

If $${\left\{\left(\frac{\sec \theta -1}{\sec \theta +1}\right)\right\}}^n=\text{cosec}\theta -\cot \theta ,$$       then n = ?

• (A) 1
• (B) 0.5
• (C) -1
• (D) -0.5

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 $$\eqalign{ & {\left\{ {\left( {\frac{{\sec \theta - 1}}{{\sec \theta + 1}}} \right)} \right\}^n} = {\text{cosec}}\,\theta - \cot \theta \cr & {\text{Put }}\theta = {45^ \circ } \cr & {\text{So}},\,{\left\{ {\left( {\frac{{\sec {{45}^ \circ } - 1}}{{\sec {{45}^ \circ } + 1}}} \right)} \right\}^n} = {\text{cosec}}\,{45^ \circ } - \cot {45^ \circ } \cr & \Rightarrow {\left\{ {\left( {\frac{{\sqrt 2 - 1}}{{\sqrt 2 + 1}}} \right)} \right\}^n} = \sqrt 2 - 1 \cr & {\text{Rationalize internally on left side, we get}} \cr & \Rightarrow {\left\{ {{{\left( {\sqrt 2 - 1} \right)}^2}} \right\}^n} = \sqrt 2 - 1 \cr & {\text{To equate }}n{\text{ should be }}\frac{1}{2} \cr & {\text{So, }}n = 0.5 \cr} $$

349.

If $$\theta +\varphi =\frac{\pi }{2}$$   and $$\sin \theta =\frac{1}{2},$$   then the value of $$\text{sin}\varphi$$  is?

• (A) 1
• (B) $$\frac{1}{\sqrt{2}}$$
• (C) $$\frac{1}{2}$$
• (D) $$\frac{\sqrt{3}}{2}$$

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 $$\eqalign{ & \theta + \phi = \frac{\pi }{2} \cr & \theta + \phi = {90^ \circ }\,......(i) \cr & \sin \theta = \frac{1}{2} \cr & \sin \theta = {\text{sin 3}}{0^ \circ } = \frac{1}{2}\,......(ii) \cr & {\text{Put }}\theta = {30^ \circ }{\text{ in equation (i)}} \cr & {30^ \circ } + \phi = {90^ \circ } \cr & \phi = {60^ \circ } \cr & \sin \phi = \sin {60^ \circ } = \frac{{\sqrt 3 }}{2} \cr} $$

350.

If rsinθ = 1, rcosθ = $$\sqrt{3}\text{,}$$  then the value of r²tanθ is?

• (A) 4
• (B) $$\frac{1}{\sqrt{3}}$$
• (C) $$\frac{4}{\sqrt{3}}$$
• (D) $$\text{4}\sqrt{3}$$

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 $$\eqalign{ & r\sin \theta = 1,{\text{ }}r\cos \theta = \sqrt 3 \cr & {\text{Put }}\theta = {30^ \circ } \cr & r = 2 \cr & {\text{So}},{\text{ }}{r^2}\tan \theta \cr & = {\left( 2 \right)^2} \times {\text{tan}}{30^ \circ } \cr & = 4 \times \frac{1}{{\sqrt 3 }} \cr & = \frac{4}{{\sqrt 3 }} \cr} $$