Chapter 7: Q. 105 (page 498)

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

Short Answer

Expert verified

To show that $\sin α = \frac{2 z}{1 + z^{2}}$ , first replace z with $\tan \frac{α}{2}$ 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,

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

Replacezwith $\tan \frac{α}{2}$ ,

Take RHS,

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

Step 2. Rewrite the expression.

On rewriting the expression, we get,

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

Step 3. Use double-angle formula.

Using the double-angle formula,

$\sin (2 θ) = 2 \sin θ \cos θ$

From the above equation,

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

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Rewrite the expression.

Step 3. Use double-angle formula.

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If z=tanα2, show that sinα=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

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Study anywhere. Anytime. Across all devices.

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