How do you find the exact value of sin 105 degrees?

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How do you find the exact value of sin 105 degrees?

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Answer 1 · 2015-10-25T03:32:09

Find exact value of sin (105)

Ans: $(\frac{\sqrt{2 + \sqrt{3}}}{2})$ $(sqrt(2 + sqrt3)/2)$

Explanation:

sin (105) = sin (15 + 90) = cos 15.
First find (cos 15). Call cos 15 = cos x
Apply the trig identity: $cos 2 x = 2 {cos}^{2} x - 1 .$ $cos 2x = 2cos^2 x - 1.$
cos 2x = cos (30) $= \frac{\sqrt{3}}{2} = 2 {cos}^{2} x - 1$ $= sqrt3/2 = 2cos^2 x - 1$
2cos^2 x = 1 + sqrt3/2 = (2 + sqrt3)/2
cos^2 x = (2 + sqrt3)/4
cos x = cos 15 = (sqrt(2 + sqrt3)/2. (since cos 15 is positive)

$sin (105) = cos (15) = \frac{\sqrt{2 + \sqrt{3}}}{2} .$ $sin (105) = cos (15) = sqrt(2 + sqrt3)/2.$
Check by calculator.
sin (105) = cos 15 = 0.97
$\frac{\sqrt{2 + \sqrt{3}}}{2} = \frac{1.93}{2} = 0.97 .$ $sqrt(2 + sqrt3)/2 = 1.93/2 = 0.97.$ OK

Answer 2 · 2016-02-10T12:31:02

Use $sin 105 = sin (60 + 45) = sin 60 cos 45 + cos 60 sin 45$ $sin 105 = sin (60 + 45) = sin 60 cos 45 + cos 60 sin 45$
$= (\frac{\sqrt{3}}{2}) (\frac{1}{\sqrt{2}}) + (\frac{1}{2}) (\frac{1}{\sqrt{2}}) = \frac{\sqrt{2}}{4} ((\sqrt{3} + 1) = 0.9656$ $=(sqrt3/2)(1/sqrt2)+(1/2)(1/sqrt2)=sqrt2/4((sqrt3+1)=0.9656$ nearly.

Explanation:

sin 45, cos 45 and sin 60 are irrational. So, the answer is a surd.

Answer 3 · 2017-01-29T06:40:14

${sin 105}^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}$ $sin105^@=(sqrt6+sqrt2)/4$

Explanation:

We know $sin (A + B) = sin A cos B + cos A sin B$ $sin(A+B)=sinAcosB+cosAsinB$

Hence ${sin 105}^{\circ}$ $sin105^@$

= $sin (60^{\circ} + 45^{\circ})$ $sin(60^@+45^@)$

= ${sin 60}^{\circ} {cos 45}^{\circ} + {cos 60}^{\circ} {sin 45}^{\circ}$ $sin60^@cos45^@+cos60^@sin45^@$

= $\frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}} + \frac{1}{2} \times \frac{1}{\sqrt{2}}$ $sqrt3/2xx1/sqrt2+1/2xx1/sqrt2$

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

= $\frac{\sqrt{6} + \sqrt{2}}{4}$ $(sqrt6+sqrt2)/4$

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How do you find the exact value of sin 105 degrees?

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