Question

What is the value of $\sin \dfrac{\pi }{4}?$

Answer 1

Hint: We will draw a right triangle with two of the angles equal to $45{}^\circ .$ We need to find the length of the hypotenuse of the triangle when the lengths of the other two sides of the triangle are equal to $1.$ Then we will use the trigonometry to find the hypotenuse. Then these values will be equated.

Complete step by step solution:
Let us consider the given trigonometric function $\sin \dfrac{\pi }{4}.$
We are asked to find the value of the given trigonometric function.
Before starting, let us convert the angle from the radian measure to the degree measure.
We know that ${{1}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}.$
So, to convert the angle value from the radian measure to the degree measure, we need to multiply the value in radian measure with $\dfrac{180}{\pi }.$
Therefore, the given angle $\dfrac{\pi }{4}=\dfrac{\pi }{4}\dfrac{180}{\pi }.$
We will get $\dfrac{\pi }{4}={{45}^{\circ }}.$
Now, we will draw a right triangle with height=base =$1$ unit and the two equal angles equal to ${{45}^{\circ }}.$
Then, we will use Pythagoras theorem to find the hypotenuse.
We will get, if $a$ is the eight and $b$ is the base length, then the hypotenuse $=\sqrt{{{a}^{2}}+{{b}^{2}}}.$

Therefore, the hypotenuse of our triangle will be equal to $\sqrt{{{1}^{2}}+{{1}^{2}}}=\sqrt{2}$ units.
We know that the trigonometric Sine function is defined as $\sin \theta =\dfrac{opposite\ side}{Hypotenuse}.$
We will take $\theta ={{45}^{\circ }}.$ Since two of the angles other than the right angle is ${{45}^{\circ }},$ we can consider any of these angles. Since the sides are also equal, the value will not change.
Therefore, $\sin {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}.$
Hence the value of $\sin \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}.$

Note: In the similar way, we can find the value of $\cos \dfrac{\pi }{4}.$ The Cosine function is defined as $\cos \theta =\dfrac{Adjacent\ side}{Hypotenuse}.$ We will get $\cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}.$ The Tangent function is defined as $\tan \theta =\dfrac{\sin \theta }{\cos \theta }.$ We will get $\tan \dfrac{\pi }{4}=1.$