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Question: How do you evaluate \( {\sin ^2}(\dfrac{\pi }{4}) \) ?...

How do you evaluate sin⁡2(π4){\sin ^2}(\dfrac{\pi }{4}) ?

Explanation

Solution

Hint : We calculate the values of the common angles between 0 degrees and 90 degrees by using an equilateral triangle or by a unit circle, so we know the value of sin⁡π4\sin \dfrac{\pi }{4} . In this question, we have to find the square of sin⁡π4\sin \dfrac{\pi }{4} , that is, we have to find the value obtained on multiplying sin⁡π4\sin \dfrac{\pi }{4} with itself. So, putting the known value and then multiplying it with itself will give us the correct answer.

Complete step-by-step answer :
Trigonometric ratios tell us the relation between the sides of a right-angled triangle; the main functions of trigonometry are sine, cosine and tangent functions, sine function is the ratio of the perpendicular and the hypotenuse of the right-angled triangle, cosine function is the ratio of the base and the hypotenuse of the right-angled triangle and tangent is the ratio of the sine function and the cosine function. Cosecant, secant and cotangent are the reciprocal of sine, cosine and tangent functions.
Here we have been given,
sin⁡2(π4){\sin ^2}(\dfrac{\pi }{4})
sin⁡π4=12\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}
On squaring both the sides of the above equation, we get –
sin⁡2(π4)=12×12 ⇒sin⁡2(π4)=12   {\sin ^2}(\dfrac{\pi }{4}) = \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }} \\\ \Rightarrow {\sin ^2}(\dfrac{\pi }{4}) = \dfrac{1}{2} \;
Hence, the value of sin⁡2(π4){\sin ^2}(\dfrac{\pi }{4}) is 12\dfrac{1}{2} .
So, the correct answer is “12\dfrac{1}{2} ”.

Note : There are two ways to express the measure of angles, one is radians and the other is degrees, in this question, the angle is given in radians. The angles can be converted from one form to another by a simple method. So, if we remember the measure of trigonometric angles in the form of degrees, we can convert it into radians and then find out the answer.