How do you use half angle identities to solve equations?

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How do you use half angle identities to solve equations?

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Answer 1 · 2015-04-12T14:06:48

Common Half angle identity:
1. $sin a = 2 sin (\frac{a}{2}) \cdot cos (\frac{a}{2})$ $sin a = 2 sin (a/2)* cos (a/2)$

Half angle Identities in term of t = tan a/2.
2. $sin a = \frac{2 t}{1 + t^{2}}$ $sin a = (2t)/(1 + t^2)$

3. $cos a = \frac{1 - t^{2}}{1 + t^{2}}$ $cos a = (1 - t^2)/(1 + t^2)$

$tan a = \frac{2 t}{1 - t^{2}} .$ $tan a = (2t)/(1 - t^2).$

Use of half angle identities to solve trig equations.

Example. Solve $cos x + 2 \cdot sin x = 1 + tan (\frac{x}{2}) .$ $cos x + 2*sin x = 1 + tan (x/2).$
Solution. Call $t = tan (\frac{x}{2})$ $t = tan (x/2)$ . Use half angle identities (2) and (3) to transform the equation.

$\frac{1 - t^{2}}{4} + \frac{1 + t^{2}}{4} = 1 + t .$ $(1 - t^2)/4 + (1 + t^2)/4 = 1 + t.$

$1 - t^{2} + 4 t = (1 + t) (1 + t^{2})$ $1 - t^2 + 4t = (1 + t)(1 + t^2)$

$t^{3} + 2 t^{2} - 3 t = t \cdot (t^{2} + 2 t - 3) = 0 .$ $t^3 + 2t^2 - 3t = t*(t^2 + 2t - 3) = 0.$

Next, solve the $3$ $3$ basic trig equations: $tan (\frac{x}{2}) = t = 0; tan (\frac{x}{2}) = - 3;$ $tan (x/2) = t = 0; tan (x/2) = -3;$ and $tan (\frac{x}{2}) = 1 .$ $tan (x/2) = 1.$

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How do you use half angle identities to solve equations?

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