SATHEE: Trigonometrical Equations 3 Question 20

Trigonometrical Equations 3 Question 20

21. Prove that $5 \cos θ + 3 \cos θ + \frac{π}{3} + 3$ lies between -4 and 10.

(1979, 3M)

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Solution:

Let $f (θ) = 5 \cos θ + 3 \cos θ + \frac{π}{3} + 3$

$= 5 \cos θ + 3 \cos θ \cos \frac{π}{3} - \sin θ \sin \frac{π}{3} + 3$

$\begin{aligned} = 5 \cos θ + 3 \frac{1}{2} \cos θ - 3 \frac{\sqrt{3}}{2} \sin θ + 3 \\ = \frac{13}{2} \cos θ - \frac{3 \sqrt{3}}{2} \sin θ + 3 \\ \Rightarrow f (θ) & = \frac{1}{2} (13 \cos θ - 3 \sqrt{3} \sin θ) + 3 \end{aligned}$

Put $r \cos α = 13, r \sin α = 3 \sqrt{3}$ , then

$\begin{aligned} r & = \sqrt{169 + 27} \\ = \sqrt{196} = 14 \\ ∴ f (θ) & = \frac{1}{2} (r \cos α \cos θ - r \sin α \sin θ) + 3 \\ = \frac{1}{2} r \cos (θ + α) + 3 \\ = 7 \cos (θ + α) + 3 \end{aligned}$

Now, $- 1 \leq \cos (θ + α) \leq 1$

$\Rightarrow - 7 \leq 7 \cos (θ + α) \leq 7$

$\Rightarrow - 4 \leq f (θ) \leq 10$

Trigonometrical Equations 3 Question 20

21. Prove that $5 \cos θ + 3 \cos θ + \frac{π}{3} + 3$ lies between -4 and 10.

Solution:

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Trigonometrical Equations 3 Question 20

21. Prove that 5cos⁡θ+3cos⁡θ+π3+3 lies between -4 and 10.

Solution:

Engineering Preparation

Medical Preparation

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Other Resources

Syllabus

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Sample Mock Test

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NCERT Exercise Video Solution

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21. Prove that $5 \cos θ + 3 \cos θ + \frac{π}{3} + 3$ lies between -4 and 10.