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Question: How do you evaluate \(\sin (\dfrac{\pi }{6})\)...

How do you evaluate sin⁡(π6)\sin (\dfrac{\pi }{6})

Explanation

Solution

This sum can be solved directly as it is a common value of sin⁡\sin function. In order to solve such a type of sum with values not common for example sin⁡(120)\sin (120)a student has to make use of the formula of sin⁡\sin to evaluate. To evaluate the above sum the student can substitute the value directly or can use the formula sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A\cos B - \cos A\sin B as we know the value of sin⁡(90)&sin⁡(60)\sin (90)\& \sin (60). Another way to solve such a type of sum is by proving it. A student can take an equilateral triangle and use the properties of hypotenuse and similarity of triangles to evaluate sin⁡(π6)\sin (\dfrac{\pi }{6}).

Complete step-by-step answer:
1st Method is directly using the value and noting down the answer. Since π\pi stands for 180∘{180^ \circ } and it is divided by 66, sin⁡(π6)\sin (\dfrac{\pi }{6}) is equivalent to sin⁡(30∘)\sin ({30^ \circ }).
⇒sin⁡(30∘)=12\Rightarrow \sin ({30^ \circ }) = \dfrac{1}{2}
2nd Method is to use the formula sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A\cos B - \cos A\sin B and find the answer for sin⁡(30∘)\sin ({30^ \circ }).
We can write sin⁡(30∘)\sin ({30^ \circ }) as sin⁡(90∘−60∘)\sin ({90^ \circ } - {60^ \circ }).
Using the formula :
sin⁡(90∘−60∘)=sin⁡90×cos⁡60−sin⁡60×cos⁡90..........(1)\sin ({90^ \circ } - {60^ \circ }) = \sin 90 \times \cos 60 - \sin 60 \times \cos 90..........(1)
As we know that value of cos⁡90\cos 90 is 00,
sin⁡(90∘−60∘)=sin⁡90×cos⁡60..........(2)\sin ({90^ \circ } - {60^ \circ }) = \sin 90 \times \cos 60..........(2)
Making use of the standard values , we get the value of sin⁡(30∘)\sin ({30^ \circ })
sin⁡(90∘−60∘)=1×12.........(3)\sin ({90^ \circ } - {60^ \circ }) = 1 \times \dfrac{1}{2}.........(3)
⇒sin⁡(30∘)=12\Rightarrow \sin ({30^ \circ }) = \dfrac{1}{2}

Note: In order to solve trigonometric numerical the student should know the standard values of all the trigonometric functions which are 0∘,30∘,45∘,60∘,90∘{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ },{90^ \circ }. These values would not only be useful for trigonometric related problems but also help in calculating the values of sides in triangles, useful in heights and distances chapter. Also they are useful while solving chapters like integration, derivatives.