How do you find the exact values of the sine, cosine, and tangent of the angle $\frac{5 π}{12}$ $(5pi)/12$ ?

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How do you find the exact values of the sine, cosine, and tangent of the angle $\frac{5 π}{12}$ $(5pi)/12$ ?

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Answer 1 · 2018-03-27T08:53:12

$see explanation$ $"see explanation"$

Explanation:

$using the trigonometric identities$ $"using the "color(blue)"trigonometric identities"$

$∙ x sin (x + y) = sin x cos y + cos x sin y$ $•color(white)(x)sin(x+y)=sinxcosy+cosxsiny$

$∙ x cos (x + y) = cos x cos y - sin x sin y$ $•color(white)(x)cos(x+y)=cosxcosy-sinxsiny$

$note that \frac{5 π}{12} = \frac{π}{4} + \frac{π}{6}$ $"note that "(5pi)/12=pi/4+pi/6$

$\Rightarrow sin (\frac{5 π}{12}) = sin (\frac{π}{4} + \frac{π}{6})$ $rArrsin((5pi)/12)=sin(pi/4+pi/6)$

$\Rightarrow sin (\frac{π}{4} + \frac{π}{6})$ $rArrsin(pi/4+pi/6)$

$= sin (\frac{π}{4}) cos (\frac{π}{6}) + cos (\frac{π}{4}) sin (\frac{π}{6})$ $=sin(pi/4)cos(pi/6)+cos(pi/4)sin(pi/6)$

$= (\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}) + (\frac{1}{\sqrt{2}} \times \frac{1}{2})$ $=(1/sqrt2xxsqrt3/2)+(1/sqrt2xx1/2)$

$= \frac{\sqrt{3}}{2 \sqrt{2}} + \frac{1}{2 \sqrt{2}}$ $=sqrt3/(2sqrt2)+1/(2sqrt2)$

$= \frac{\sqrt{3} + 1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{1}{4} (\sqrt{6} + \sqrt{2}) \leftarrow exact value$ $=(sqrt3+1)/(2sqrt2)xxsqrt2/sqrt2=1/4(sqrt6+sqrt2)larrcolor(red)"exact value"$

$cos (\frac{5 π}{12}) = cos (\frac{π}{4} + \frac{π}{6})$ $cos((5pi)/12)=cos(pi/4+pi/6)$

$\Rightarrow cos (\frac{π}{4} + \frac{π}{6})$ $rArrcos(pi/4+pi/6)$

$= cos (\frac{π}{4}) cos (\frac{π}{6}) - sin (\frac{π}{4}) sin (\frac{π}{6})$ $=cos(pi/4)cos(pi/6)-sin(pi/4)sin(pi/6)$

$= (\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}) - (\frac{1}{\sqrt{2}} \times \frac{1}{2})$ $=(1/sqrt2xxsqrt3/2)-(1/sqrt2xx1/2)$

$= \frac{\sqrt{3} - 1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ $=(sqrt3-1)/(2sqrt2)xxsqrt2/sqrt2$

$= \frac{1}{4} (\sqrt{6} - \sqrt{2}) \leftarrow exact value$ $=1/4(sqrt6-sqrt2)larrcolor(red)"exact value"$

$tan (\frac{5 π}{12}) = \frac{sin (\frac{5 π}{12})}{cos (\frac{5 π}{12})}$ $tan((5pi)/12)=sin((5pi)/12)/cos((5pi)/12)$

$\times \times \times x = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} \times \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ $color(white)(xxxxxxx)=(sqrt6+sqrt2)/(sqrt6-sqrt2)xx(sqrt6+sqrt2)/(sqrt6+sqrt2)$

$\times \times \times x = \frac{6 + 2 \sqrt{12} + 2}{4}$ $color(white)(xxxxxxx)=(6+2sqrt12+2)/4$

$\times \times \times x = \frac{8 + 4 \sqrt{3}}{4}$ $color(white)(xxxxxxx)=(8+4sqrt3)/4$

$\times \times \times x = 2 + \sqrt{3} \leftarrow exact value$ $color(white)(xxxxxxx)=2+sqrt3larrcolor(red)"exact value"$

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How do you find the exact values of the sine, cosine, and tangent of the angle $\frac{5 π}{12}$ $(5pi)/12$ ?

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How do you find the exact values of the sine, cosine, and tangent of the angle 5π12(5pi)/12?

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How do you find the exact values of the sine, cosine, and tangent of the angle $\frac{5 π}{12}$ $(5pi)/12$ ?