Q. 111 Show that聽sin3胃+sin3胃+120掳+s... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 111 (page 488)

Show that $\sin^{3} 胃 + \sin^{3} (胃 + 120 掳) + \sin^{3} (胃 + 240 掳) = - \frac{3}{4} \sin (3 胃)$

Short Answer

Expert verified

To show that $\sin^{3} 胃 + \sin^{3} (胃 + 120 掳) + \sin^{3} (胃 + 240 掳) = - \frac{3}{4} \sin (3 胃)$ , take the LHS and simplify using the identity formulas.

Step by step solution

Step 1. Given information.

Consider the given question,

$\sin^{3} 胃 + \sin^{3} (胃 + 120 掳) + \sin^{3} (胃 + 240 掳) = - \frac{3}{4} \sin (3 胃)$

Take the LHS,

localid="1646423145211" $= \sin^{3} 胃 + \sin^{3} (胃 + 120 掳) + \sin^{3} (胃 + 240 掳) = \sin^{3} 胃 + {(\sin 胃 \cos 120 掳 + \cos 胃 \sin 120 掳)}^{3} + {(\sin 胃 \cos 240 掳 + \cos 胃 \sin 240 掳)}^{3} = \sin^{3} 胃 + {(- \frac{1}{2} \sin 胃 + \frac{\sqrt{3}}{2} \cos 胃)}^{3} + {(- \frac{1}{2} \sin 胃 - \frac{\sqrt{3}}{2} \cos 胃)}^{3} = \sin^{3} 胃 + (- \sin 胃) (\frac{1}{4} \sin^{2} 胃 + \frac{3}{4} \cos^{2} 胃 - \frac{1}{4} \sin^{2} 胃 + \frac{3}{4} \cos^{2} 胃)$

Step 2. Use the identity formulas and simplify the expression.

Continuing the above expression,

$= \frac{3}{4} \sin^{3} 胃 - \frac{9}{4} \cos^{2} 胃 \sin 胃 = \sin^{3} 胃 + (- \frac{\sin^{3} 胃}{4} - \frac{9}{4} \cos^{2} 胃 \sin 胃)$

Use the identity $\cos^{2} 胃 = 1 - \sin^{2} 胃$ ,

= \frac{3}{4} \{\sin^{3} 胃 - 3 \sin 胃 (1 - \sin^{2} 胃)\} = \frac{3}{4} (4 \sin^{3} 胃 - 3 \sin 胃) = - \frac{3}{4} \sin (3 胃)

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91影视

Show that $\sin^{3} 胃 + \sin^{3} (胃 + 120 掳) + \sin^{3} (胃 + 240 掳) = - \frac{3}{4} \sin (3 胃)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the identity formulas and simplify the expression.

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91影视

Show thatsin3胃+sin3胃+120掳+sin3胃+240掳=-34sin3胃

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the identity formulas and simplify the expression.

One App. One Place for Learning.

Most popular questions from this chapter

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Show that $\sin^{3} 胃 + \sin^{3} (胃 + 120 掳) + \sin^{3} (胃 + 240 掳) = - \frac{3}{4} \sin (3 胃)$