Question

What is the value of: $\sin {{0}^{c}}+2\cos {{0}^{c}}+3\sin {{\left( \dfrac{\pi }{2} \right)}^{c}}+4\cos {{\left( \dfrac{\pi }{2} \right)}^{c}}+5\sec {{0}^{c}}+6\cos ec{{\left( \dfrac{\pi }{2} \right)}^{c}}$?

Answer 1

Hint: Put the trigonometric values of the given angles of trigonometric functions and simplify the addition. Angles can also be converted into degrees by using the formula: $\theta \text{ rad}=\left( \dfrac{180}{\pi }\times \theta \right)\text{degrees}$.

Complete step by step answer:

One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius of the circle. In general, mathematically, $\theta =\dfrac{l}{r}$. As it is a ratio of two lengths, hence, it is a pure number.
Now, let us come to the question. We have been given the expression: $\sin {{0}^{c}}+2\cos {{0}^{c}}+3\sin {{\left( \dfrac{\pi }{2} \right)}^{c}}+4\cos {{\left( \dfrac{\pi }{2} \right)}^{c}}+5\sec {{0}^{c}}+6\cos ec{{\left( \dfrac{\pi }{2} \right)}^{c}}$
On converting these radians into degrees, we get,
\[{{0}^{c}}=\dfrac{180}{\pi }\times 0=\text{ 0 degrees, and }{{\left( \dfrac{\pi }{2} \right)}^{c}}=\dfrac{180}{\pi }\times \dfrac{\pi }{2}=\text{ }{{90}^{\circ }}\text{ degrees}\].
Therefore, the expression becomes
$\sin {{0}^{\circ }}+2\cos {{0}^{\circ }}+3\sin {{90}^{\circ }}+4\cos {{90}^{\circ }}+5\sec {{0}^{\circ }}+6\cos ec{{90}^{\circ }}$
We know that, \[\sin {{0}^{\circ }}=\cos {{90}^{\circ }}=0\text{ and }\cos {{0}^{\circ }}=\sin {{90}^{\circ }}=1\]. Therefore substituting these values in the expression, we get,
$\begin{align}
  & \sin {{0}^{\circ }}+2\cos {{0}^{\circ }}+3\sin {{90}^{\circ }}+4\cos {{90}^{\circ }}+5\sec {{0}^{\circ }}+6\cos ec{{90}^{\circ }} \\
 & =0+2+3+0+5+6 \\
 & =16 \\
\end{align}$
Hence, the value of the given expression is 16.

Note: Generally we will come across $\theta $ in radians and not in degrees in higher classes, so it will be helpful to us if we will remember the values of trigonometric functions of some particular angles given in radians. Here, we were not required to convert the angles in degrees as it takes more steps and time to solve the question.