Q. 105 If z=tan伪2, show that sin伪=2z1... [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. 105 (page 498)

If $z = \tan \frac{伪}{2}$ , show that $\sin 伪 = \frac{2 z}{1 + z^{2}}$ .

Short Answer

Expert verified

To show that $\sin 伪 = \frac{2 z}{1 + z^{2}}$ , first replace z with $\tan \frac{伪}{2}$ in RHS and rewrite the expression. Then use the double-angle formula and simplify the equation.

Step by step solution

Step 1. Given information.

Consider the given question,

$z = \tan \frac{伪}{2}, \sin 伪 = \frac{2 z}{1 + z^{2}}$

Replacezwith $\tan \frac{伪}{2}$ ,

Take RHS,

$\frac{2 z}{1 + z^{2}} = \frac{2 \tan (\frac{伪}{2})}{1 + \tan^{2} (\frac{伪}{2})} = \frac{2 \tan (\frac{伪}{2})}{s e c^{2} (\frac{伪}{2})}$

Step 2. Rewrite the expression.

On rewriting the expression, we get,

$\frac{2 \tan (\frac{伪}{2})}{s e c^{2} (\frac{伪}{2})} = \frac{2 \frac{\sin (\frac{伪}{2})}{\cos (\frac{伪}{2})}}{\frac{1}{\cos^{2} (\frac{伪}{2})}} = 2 \frac{\sin (\frac{伪}{2})}{\cos (\frac{伪}{2})} 脳 \cos^{2} (\frac{伪}{2}) = 2 \sin (\frac{伪}{2}) \cos^{2} (\frac{伪}{2})$

Step 3. Use double-angle formula.

Using the double-angle formula,

$\sin (2 胃) = 2 \sin 胃 \cos 胃$

From the above equation,

role="math" localid="1646415465820" $= 2 \sin (\frac{伪}{2}) \cos (\frac{伪}{2}) = \sin (2 脳 \frac{伪}{2}) = \sin 伪$

Therefore, $L H S = R H S$ .

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91影视

If $z = \tan \frac{伪}{2}$ , show that $\sin 伪 = \frac{2 z}{1 + z^{2}}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Rewrite the expression.

Step 3. Use double-angle formula.

One App. One Place for Learning.

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91影视

If z=tan伪2, show that sin伪=2z1+z2.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Rewrite the expression.

Step 3. Use double-angle formula.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

If $z = \tan \frac{伪}{2}$ , show that $\sin 伪 = \frac{2 z}{1 + z^{2}}$ .