Question

How do you evaluate sine, cosine, tangent of $750^\circ$ without using a calculator?

Answer 1

Hint: Given an angle in degrees and we have to find the sine, cosine, and tangent of a particular measure without the help of the calculator. First, we will convert the given angle between the angle of value $0$ to $90^\circ $. Then, we will substitute the value of the trigonometric function for the measure of an angle. Then, compute the exact value of the required angle.
Formula used:
The reference angle for the angle $\theta $ is given by:
$\theta + n \left( 360^\circ \right) $

Complete step by step solution:
We are given the measure of an angle in degrees. First, we will find the reference angle to convert the given angle between$0$ to $90^\circ $.
For this, we will divide $750^\circ $ by $360^\circ $.
$ \Rightarrow 750^\circ \div 360^\circ = 2.08$
This means two cycles fit within the angle.
$ \Rightarrow 360^\circ \times 2 = 720^\circ $
Now, the angle which is left over $ \Rightarrow 750^\circ - 720^\circ = 30^\circ $
Thus, the reference angle is $30^\circ $
Therefore, the value of sine $750^\circ $ is equal to sine $30^\circ $
$ \Rightarrow \sin 750^\circ = \sin 30^\circ $
Substitute the value of $\sin 30^\circ = \dfrac{1}{2}$
$ \Rightarrow \sin 750^\circ = \dfrac{1}{2}$
$ \Rightarrow \sin 750^\circ = 0.5$
Similarly we will find the value of cosine $750^\circ $.
$ \Rightarrow \cos 750^\circ = \cos 30^\circ $
Substitute the value of $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$
$ \Rightarrow \cos 750^\circ = \dfrac{{\sqrt 3 }}{2}$
Now, simplify the expression.
$ \Rightarrow \cos 750^\circ = 0.866$
Now, we will find the value of tan $750^\circ $ using the trigonometric identity, $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
$ \Rightarrow \tan 750^\circ = \dfrac{{\sin 750^\circ }}{{\cos 750^\circ }}$
Now, substitute the values into the expression.
$ \Rightarrow \tan 750^\circ = \dfrac{{0.5}}{{0.866}}$
$ \Rightarrow \tan 750^\circ = 0.577$
Hence, the value of sine, cosine and tangent of $750^\circ $ is $0.5$, $0.866$ and $0.577$ respectively.

Note: Students must remember that we have used the trigonometric values of a particular measure of an angle. Here, the value of $\sin 30^\circ = \dfrac{1}{2}$ and the value of $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$. Students may note that we will find the reference angle by dividing the given angle by $2\pi $.