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Question: How do you find the exact value of \({(\sin 30)^2} + {(\cos 30)^2}\)?...

How do you find the exact value of (sin⁡30)2+(cos⁡30)2{(\sin 30)^2} + {(\cos 30)^2}?

Explanation

Solution

For solving this particular question , you have to simplify the given expression by reordering the equation , applying trigonometric identity , taking square root. To find the exact value of (sin⁡30)2+(cos⁡30)2{(\sin 30)^2} + {(\cos 30)^2} , we must know that sin⁡30=12\sin 30 = \dfrac{1}{2} and cos⁡30=32\cos 30 = \dfrac{{\sqrt 3 }}{2} .

Complete step by step solution:
We have to find the exact value of
(sin⁡30)2+(cos⁡30)2............(A){(\sin 30)^2} + {(\cos 30)^2}............(A)
We know that sin⁡30=12...........(1)\sin 30 = \dfrac{1}{2}...........(1) and cos⁡30=32.........(2)\cos 30 = \dfrac{{\sqrt 3 }}{2}.........(2) .
Now, substitute the (1),(2)(1),(2) in the equation (A)(A) ,
⇒(sin⁡30)2+(cos⁡30)2 =(12)2+(32)2 =44 =1  \Rightarrow {(\sin 30)^2} + {(\cos 30)^2} \\\ = {\left( {\dfrac{1}{2}} \right)^2} + {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} \\\ = \dfrac{4}{4} \\\ = 1 \\\
Therefore , the exact value of (sin⁡30)2+(cos⁡30)2{(\sin 30)^2} + {(\cos 30)^2} is 11 .

Additional Information:
In arithmetic, pure mathematics identities are equalities that involve pure mathematics functions and are true for every worth of the occurring variables that every aspect of the equality is outlined. Geometrically, these are identities involving sure functions of one or additional angles. We have a trigonometry formula which says sin⁡θ=ph\sin \theta = \dfrac{p}{h} , where pp represent length of perpendicular side and hhrepresent length of hypotenuse side. We have another trigonometry formula which says cos⁡θ=bh\cos \theta = \dfrac{b}{h} , where bb represents length of base and hh represent length of hypotenuse side.

Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. In order to solve and simplify the given expression we can also use the identity that is sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 and express our given expression in the simplest form and thereby solve it. Identities are helpful whenever expressions involving pure mathematics functions should be simplified. An important application is that the combination of non-trigonometric functions: a typical technique involves initial mistreatment the substitution rule with a mathematical relation, then simplifying the ensuing integral with a pure mathematics identity.