Question

Find the values of the trigonometric function $\sin \left( {{765}^{0}} \right)$

Answer 1

Hint: In this question, first work out on the conversion of the degree to radian by using the formula \[{{1}^{0}}={{\left( \dfrac{\pi }{180} \right)}^{c}}\]. Second work out on the value of the given trigonometric function by using the formula $\sin \left( 2n\pi +\theta \right)=\sin \theta $.

 Complete step-by-step answer:
To convert degrees to radians, take the number of degrees to be converted and multiply it by $\dfrac{\pi }{180}$ . You can calculate this by converting both numbers into fractions. For example, to convert 120 degrees you would have $120\times \dfrac{\pi }{180}=\dfrac{120\pi }{180}$ .

The given angle in degree is $765=\dfrac{\pi }{180}\times 765=\dfrac{17\pi }{4}$ in radian.
The given trigonometric function is

$\sin \left( {{765}^{0}} \right)=\sin \left( \dfrac{17\pi }{4} \right)$

Rearranging the value of the angle, we get

$\sin \left( {{765}^{0}} \right)=\sin \left( 4\pi +\dfrac{\pi }{4} \right)$

The trigonometric ratio of an angle does not change if it is increased or decreased by a

multiple of $2\pi $ . Hence $\sin \left( 2n\pi +\theta \right)=\sin \theta $

$\sin \left( {{765}^{0}} \right)=\sin \left( \dfrac{\pi }{4} \right)$

We know that the value of $\sin \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}$

$\sin \left( {{765}^{0}} \right)=\dfrac{1}{\sqrt{2}}$

Hence, the value of the trigonometric function $\sin \left( {{765}^{0}} \right)$ is $\dfrac{1}{\sqrt{2}}$



Note: You might get confused about the difference between degree measure and Radian measure. Degrees measure angles by how far you tilted our heads. Radians measure angles by distance traveled or angle in radians ($\theta $) is arc length (s) divided by radius (r). The degree measure and radian measure are denoted by ${{x}^{0}}$ and ${{x}^{C}}$ respectively.