SATHEE: Trigonometrical Ratios and Identities 1 Question 10

Trigonometrical Ratios and Identities 1 Question 10

10. Given both $θ$ and $φ$ are acute angles and $\sin θ = \frac{1}{2}, \cos φ = \frac{1}{3}$ , then the value of $θ + φ$ belongs to

(2004, 1M)

(a) $\frac{π}{3}, \frac{π}{6}$

(b) $\frac{π}{2}, \frac{2 π}{3}$

(d) $\frac{5 π}{6}, π$

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Answer:

Correct Answer: 10. (b)

Solution:

Since, $\sin θ = \frac{1}{2}$

and $\cos φ = \frac{1}{3} \Rightarrow θ = \frac{π}{6}$ and $0 < \cos φ = \frac{1}{3} < \frac{1}{2}$ as $0 < \frac{1}{3} < \frac{1}{2}$

$\Rightarrow θ = \frac{π}{6}$ and $\cos^{- 1} (0) > φ > \cos^{- 1} \frac{1}{2}$

the sign changed as $\cos x$ is decreasing between $0, \frac{π}{2}$

$\begin{array}{ll} \Rightarrow & θ = \frac{π}{6} and \frac{π}{3} < φ < \frac{π}{2} \Rightarrow \frac{π}{2} < θ + φ < \frac{2 π}{3} \\ ∴ & θ \in \frac{π}{2}, \frac{2 π}{3} \end{array}$

Trigonometrical Ratios and Identities 1 Question 10

10. Given both $θ$ and $φ$ are acute angles and $\sin θ = \frac{1}{2}, \cos φ = \frac{1}{3}$ , then the value of $θ + φ$ belongs to

Answer:

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Trigonometrical Ratios and Identities 1 Question 10

10. Given both θ and φ are acute angles and sin⁡θ=12,cos⁡φ=13, then the value of θ+φ belongs to

Answer:

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Medical Preparation

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10. Given both $θ$ and $φ$ are acute angles and $\sin θ = \frac{1}{2}, \cos φ = \frac{1}{3}$ , then the value of $θ + φ$ belongs to