Question

How do you find the terminal point on the unit circle determined by $t = \dfrac{{5\pi }}{{12}}$ ?

Answer 1

In this question we need to find the terminal point on the unit circle determined by $t = \dfrac{{5\pi }}{{12}}$ . Here, we will use the cosine as $x$ -coordinate and the sine as $y$ -coordinate are mostly acute-angle measures. Then, we will convert the radian into degrees. At last, we will find the values and substitute it, which is the required coordinates.

Complete step-by-step solution:
Now, we need to find the terminal point on the unit circle determined by $t = \dfrac{{5\pi }}{{12}}$ .
Generally, the terminal point on the unit circle has the cosine as $x$ -coordinate and the sine as $y$ - coordinate are mostly acute-angle measures.
Here $t = \dfrac{{5\pi }}{{12}}$ , therefore for the $x$ -coordinate,
$\cos t = \dfrac{{5\pi }}{{12}}$
Then, for the $y$ -coordinate,
$\sin t = \dfrac{{5\pi }}{{12}}$
To convert radians into degrees, multiply by $180\pi$ , since a full circle is $360^\circ$ .
Therefore we have, $\cos t = \cos 75$ and $\sin t = \sin 75$ .
Then, $\cos 75 = 0.258$
And, $\sin 75 = 0.965$
$\Rightarrow \left( {\cos \dfrac{{5\pi }}{{12}},\sin \dfrac{{5\pi }}{{12}}} \right) = \left( {0.258,0.965} \right)$

Hence, the terminal point on the unit circle determined by $t = \dfrac{{5\pi }}{{12}}$ is $\left( {0.258,0.965} \right)$ .

Note: In this question, it is important to note that the unit circle is a circle with its centre at the origin of the coordinate plane and with a radius of $1\,unit$ . If $\left( {x,y} \right)$ are the coordinates of a point on the circle, then for the right-triangle the Pythagorean theorem for unit circle is ${x^2} + {y^2} = 1$ . The terminal point is the ray that has been rotated around the origin to form an angle with the stationary ray that is the initial side of the angle. They are sine and cosine values of the most common acute-angle measures. However be careful when converting the radian into degree.