JEE Maths - Trigonometric Functions

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Question - 1

If \(\sin ^{16} \alpha=\frac{1}{5}\), then the value of \(\frac{1}{\cos ^{2} \alpha}+\frac{1}{1+\sin ^{2} \alpha}+\frac{2}{1+\sin ^{4} \alpha}+\frac{4}{1+\sin ^{8} \alpha}\) is equal to ____________ .

  • A 2
  • B 4
  • C 6
  • D 10

Question - 2

If a, b, g, d are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of \(4 \sin \frac{\alpha}{2}+3 \sin \frac{\beta}{2}+2 \sin \frac{\gamma}{2}+\sin \frac{\delta}{2} \text { is }\)equal to ____________ .

  • A \(2 \sqrt{1-k} \)
  • B \(2 \sqrt{1+k} \)
  • C \(2 \sqrt{k} \)
  • D None of these

Question - 3

\(If \cos \alpha=\frac{2 \cos \beta-1}{2-\cos \beta}(0<\alpha<\beta<\pi), then \frac{\tan \alpha / 2}{\tan \beta / 2} \) is equal to __________ .

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

Question - 4

\(\text { Let } 0 \leq \alpha, \beta, \gamma, \delta \leq \pi\) where \(\beta \text { and } \gamma\) are not complementary such that 
\(2 \cos \alpha+6 \cos \beta+7 \cos \gamma+9 \cos \delta=0 and 2 \sin \alpha-6 \sin \beta+7 \sin \gamma-9 \sin \delta=0\)
\(\text { If } \frac{\cos (\alpha+\delta)}{\cos (\beta+\gamma)}=\frac{m}{n}\) where m and n are relatively prime positive numbers, then the value of (m + n) is equal to ____________ .

  • A 11
  • B 10
  • C 9
  • D 7

Question - 5

If \(\sin \theta+\sin 2 \theta+\sin 3 \theta=\sin \alpha and \cos \theta +\cos 2 \theta+\cos 3 \theta=\cos \alpha, then \theta\) is equal to ___________ .

  • A \(\alpha / 2 \)
  • B \(\alpha \)
  • C \(2 \alpha \)
  • D \(\alpha / 6\)

Question - 6

The value of the expression \( \frac{\sin 7 \alpha+6 \sin 5 \alpha+17 \sin 3 \alpha+12 \sin \alpha}{\sin 6 \alpha+5 \sin 4 \alpha+12 \sin 2 \alpha} \text { where } \alpha=\frac{\pi}{5} \) is equal to ___________ .

  • A \(\frac{\sqrt{5}-1}{4} \)
  • B \(\frac{\sqrt{5}+1}{4} \)
  • C \(\frac{\sqrt{5}+1}{2} \)
  • D \(\frac{\sqrt{5}-1}{2}\)

Question - 7

\(\frac{\sin \theta}{\cos (3 \theta)}+\frac{\sin (3 \theta)}{\cos (9 \theta)}+\frac{\sin (9 \theta)}{\cos (27 \theta)}+\frac{\sin (27 \theta)}{\cos (81 \theta)}=\)

  • A \(\frac{\sin (81 \theta)}{2 \cos (80 \theta) \cos \theta} \)
  • B \(\frac{\sin (80 \theta)}{2 \cos (81 \theta) \cos \theta} \)
  • C \(\frac{\sin (81 \theta)}{\cos (80 \theta) \cos \theta} \)
  • D \(\frac{\sin (80 \theta)}{\cos (81 \theta) \cos \theta}\)

Question - 8

\(\text { If } \operatorname{cosec} \theta=\frac{p+q}{p-q}, \text { then } \cot \left(\frac{\pi}{4}+\frac{\theta}{2}\right)=\)

  • A \(\sqrt{\frac{\mathrm{p}}{\mathrm{q}}} \)
  • B \(\sqrt{\frac{\mathrm{q}}{\mathrm{p}}} \)
  • C \(\sqrt{\mathrm{pq}} \)
  • D p q

Question - 9

\(If \sin 2 \theta+\sin 2 \phi=1 / 2 and \cos 2 \theta+\cos 2 \phi=3 / 2, then \cos ^{2}(\theta-\phi)=\)

  • A 3/8
  • B 5/8
  • C 3/4
  • D 5/4

Question - 10

\(\text { If } \mathrm{u}=(1+\cos \theta)(1+\cos 2 \theta)-\sin \theta \cdot \sin 2 \theta \mathrm{v}=\sin \theta(1+\cos 2 \theta)+\sin 2 \theta(1+\cos \theta), \text { then } \mathrm{u}^{2}+\mathrm{v}^{2}= \)

  • A \(4(1+\cos \theta)(1+\cos 2 \theta) \)
  • B \(4(1+\sin \theta)(1+\sin 2 \theta) \)
  • C \(4(1-\cos \theta)(1-\cos 2 \theta) \)
  • D \(4(1-\sin \theta)(1-\sin 2 \theta)\)
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