Question

How do you find the exact value of the sine, cosine, and tangent of the angle $\dfrac{{5\pi }}{{12}}$?

Answer 1

To find the values for an particular angle in trigonometry we have to break the angle into the smaller angle such as zero, thirty, forty five, sixty and ninety, because we know the direct value for these angles, or otherwise we have plot the graph and then see for the value.

Formulae Used:
$\Rightarrow \sin (A + B) = \sin A\cos B + \sin B\cos A$
$\Rightarrow \cos (A + B) = \cos A\cos B - \sin B\sin A$
$\Rightarrow \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$

Complete step by step solution:
The given question need to obtain the value of the trigonometric function for the given angle $\dfrac{{5\pi }}{{12}}$
The given angle is seventy five degree and needed to be solved by splitting the angle into forty five and thirty, we can write as:
$\Rightarrow \dfrac{{5\pi }}{{12}} = 75 = (45 + 30)$
Now first solving for “sin” we get:

$$\Rightarrow \sin (45 + 30) = \sin 45\cos 30 + \sin 30\cos 45(u\sin g\,\sin (A + B) = \sin A\cos B + \sin B\cos A) \\\ \Rightarrow \sin (45 + 30) = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }} = \dfrac{{\sqrt 3 }}{{2\sqrt 2 }} + \dfrac{1}{{2\sqrt 2 }} = \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }} \\\$$

Now solving for “cos” we get:

$$\Rightarrow \cos (45 + 30) = \cos 45\cos 30 - \sin 30\sin 45(u\sin g\,\cos (A + B) = \cos A\cos B - \sin B\sin A) \\\ \Rightarrow co\operatorname{s} (45 + 30) = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2} - \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }} = \dfrac{{\sqrt 3 }}{{2\sqrt 2 }} - \dfrac{1}{{2\sqrt 2 }} = \dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }} \\\$$

Now for “tangent” we have to use the relation which is:
$\Rightarrow \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
On solving we get:
$\Rightarrow \tan (45 + 30) = \dfrac{{\sin (45 + 30)}}{{\cos (45 + 30)}} = \dfrac{{\dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}}}{{\dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}}} = \dfrac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}$

Additional Information: Here after finding the value of “sin” we can also use trigonometric formulae to get the value for “cos”, the result from there would be same what we get here, only the thing is for not making any steps or to not recall any other formulae we go through these steps.

Note: The above question can also be solved by plotting the graph for every identity and marking the value for the given angle, but that process need exact plotting of graph which is very difficult on normal paper, so this process what we used here is appropriate to solve this question to get the values for the angles.