Q. 40 Complete the calculation in Exam... [FREE SOLUTION]

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Chapter 9: Q. 40 (page 721)

Complete the calculation in Example 7 by using the trigonometric identity $\sin^{2} (\frac{胃}{2}) = \frac{1}{2} (1 - \cos 胃)$ to show that ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃 = 4 \sqrt{2}$ .

Short Answer

Expert verified

The integral ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃$ is equals to $4 \sqrt{2}$

Step by step solution

Given information

The integral is ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃$

Calculation

Consider the integral, ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃$ .

The objective is to solve the integral within the given limits.

By using the trigonometric identity the integral can be written as follows.

Take the integral,

${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃 = {鈭�}_{0}^{2 蟺} \sqrt{2 \sin^{2} \frac{胃}{2}} d 胃 [since \sin^{2} \frac{胃}{2} = \frac{1}{2} \sqrt{1 - \cos 胃} 鈬� 2 \sin^{2} \frac{胃}{2} = \sqrt{1 - \cos 胃}] = \sqrt{2} {鈭�}_{0}^{2 蟺} \sqrt{\sin^{2} \frac{胃}{2}} d 胃 = \sqrt{2} {鈭�}_{0}^{2 蟺} \sin \frac{胃}{2} d 胃$

Thus,

${鈭�}_{0}^{2 蟺} \sqrt{2 \sin^{2} \frac{胃}{2}} d 胃 = \sqrt{2} {(- 2 \cos \frac{胃}{2})}_{0}^{2 蟺} [since {鈭�}_{0}^{2 蟺} \sin \frac{胃}{2} d 胃 = {(- 2 \cos \frac{胃}{2})}_{0}^{2 蟺}] = \sqrt{2} (- 2 \cos \frac{2 蟺}{2} + 2 \cos \frac{0}{2}) By applying the limits = \sqrt{2} (- 2 \cos 蟺 + 2 \cos \frac{0}{2}) = \sqrt{2} (- 2 (- 1) + 1 路 2)$

On further simplification,

${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃 = \sqrt{2} (2 + 2) = \sqrt{2} 脳 4 {鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃 = 4 \sqrt{2}$

Therefore, the integral ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃$ is equals to $4 \sqrt{2}$ .

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91影视

Complete the calculation in Example 7 by using the trigonometric identity $\sin^{2} (\frac{胃}{2}) = \frac{1}{2} (1 - \cos 胃)$ to show that ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃 = 4 \sqrt{2}$ .

Short Answer

Step by step solution

Given information

Calculation

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91影视

Complete the calculation in Example 7 by using the trigonometric identity sin2胃2=12(1-cos胃)to show that 鈭�02蟺1-cos胃d胃=42.

Short Answer

Step by step solution

Given information

Calculation

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Geometry

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Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Complete the calculation in Example 7 by using the trigonometric identity $\sin^{2} (\frac{胃}{2}) = \frac{1}{2} (1 - \cos 胃)$ to show that ${鈭�}_{0}^{2 蟺} \sqrt{1 - \cos 胃} d 胃 = 4 \sqrt{2}$ .