Question

Find the exact value, without a calculator. sin(5pi/12) = sin(5pi/6/2)

Answer 1

Answer:

$\frac{\sqrt{2 + \sqrt{3} } }{2}$

Answer 2

The exact value of $sin\frac{5\pi }{12}$ is $\frac{\sqrt{2+\sqrt{3} } }{2}$.

What is exact value of trigonometric function?

The exact values of trigonometric functions are values of trigonometric functions of certain angles that can be expressed exactly using expressions containing real numbers and roots of real numbers.

According to the given question, we have

A trigonometric function, $sin\frac{5\pi }{12}$

Now, the exact value of $sin\frac{5\pi }{12}$   is given by

$sin\frac{5\pi }{12} = sin\frac{\frac{5\pi }{6} }{2}$

⇒$sin\frac{\frac{5\pi }{6} }{2} =$±$\sqrt{\frac{1-cos\frac{5\pi }{6} }{2} }$

⇒$sin\frac{\frac{5\pi }{6} }{2}$ =±$\sqrt{\frac{1-cos(\pi -\frac{\pi }{6}) }{2} }$

⇒$sin\frac{\frac{5\pi }{6} }{2} =$ ±$\sqrt{\frac{1+cos\frac{\pi }{6} }{2} }$

⇒ $sin\frac{\frac{5\pi }{6} }{2} =$ ± $\sqrt{\frac{1+\frac{\sqrt{3} }{2} }{2} }$

⇒$sin\frac{\frac{5\pi }{6} }{2}$ =± $\sqrt{\frac{2+\sqrt{3} }{4} }$

⇒$sin\frac{\frac{5\pi }{6} }{2} = \frac{\sqrt{2+\sqrt{3} } }{2}$   (considering positive value)

Since, $sin\frac{5\pi }{12}$ lies in first quadrant and sin is positive in first quadrant. Therefore, we will consider only positive value.

Hence, the exact value of $sin\frac{5\pi }{12}$ is $\frac{\sqrt{2+\sqrt{3} } }{2}$.

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