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Question: How do you find the value of \[{\sin ^{ - 1}}(\dfrac{3}{5})\]?...

How do you find the value of sin⁡−1(35){\sin ^{ - 1}}(\dfrac{3}{5})?

Explanation

Solution

We will use the trigonometric identity cos⁡2+sin⁡2=1{\cos ^2} + {\sin ^2} = 1. We will use the Pythagoras theorem here, i.e. sin⁡θ=PH\sin \theta = \dfrac{P}{H} and cos⁡θ=BH\cos \theta = \dfrac{B}{H} where PP means perpendicular, B is base and H means hypotenuse. The inverse of sin function is denoted by Arcsine or sin⁡−1{\sin ^{ - 1}}

Complete step by step answer:
Let, sin⁡−1(35)=θ{\sin ^{ - 1}}(\dfrac{3}{5}) = \theta
⇒sin⁡θ=35\Rightarrow \sin \theta = \dfrac{3}{5}

So the value of θ\theta will be −π2⩽θ⩽π2\dfrac{{ - \pi }}{2} \leqslant \theta \leqslant \dfrac{\pi }{2}
Since, we know that, cos⁡θ=BH\cos \theta = \dfrac{B}{H}
where BB is the base and H is the hypotenuse.
From the above diagram, the value of cos will be,
cos⁡θ=45\cos \theta = \dfrac{4}{5}

Another method to solve this is by using trigonometry identity.
sin⁡θ=35\sin \theta = \dfrac{3}{5}
So, apply the trigonometry identity here i.e.
cos⁡2θ=1−sin⁡2θ{\cos ^2}\theta = 1 - {\sin ^2}\theta
Taking square root on both the sides, we get,
⇒cos⁡θ=1−sin⁡2θ\Rightarrow \cos \theta = \sqrt {1 - {{\sin }^2}\theta }
Substituting the value, we get,
⇒cos⁡θ=1−(35)2\Rightarrow \cos \theta = \sqrt {1 - {{(\dfrac{3}{5})}^2}}
Removing the brackets, we get,
⇒cos⁡θ=1−925\Rightarrow \cos \theta = \sqrt {1 - \dfrac{9}{{25}}}
Simplify the above equation, we get,
⇒cos⁡θ=25−925\Rightarrow \cos \theta = \sqrt {\dfrac{{25 - 9}}{{25}}}
⇒cos⁡θ=1625\Rightarrow \cos \theta = \sqrt {\dfrac{{16}}{{25}}}
⇒cos⁡θ=(45)2\Rightarrow \cos \theta = \sqrt {{{(\dfrac{4}{5})}^2}}
∴cos⁡θ=45\therefore \cos \theta = \dfrac{4}{5}

Note: The expression sin⁡−1(x){\sin ^{ - 1}}(x)is not the same as 1sin⁡(x)\dfrac{1}{{\sin (x)}}. In other words, −1 - 1 is not an exponent. Instead, it simply means inverse function. The trigonometric functions sinx, cosx and tanx can be used to find an unknown side length of a right triangle, if one side length and an angle measure are known. The inverse trigonometric functions sin⁡−1x,cos⁡−1x,tan⁡−1x{\sin ^{ - 1}}x,{\cos ^{^{ - 1}}}x,{\tan ^{ - 1}}x, are used to find the unknown measure of an angle of a right triangle when two side lengths are known. Pythagoras’ Theorem describes the mathematical relationship between three sides of a right-angled triangle. Trigonometry is a field of study in mathematics which observes the relationships of the sides and angles of triangles. The symbol θ\theta is used to describe an unknown angle. These functions are defined as the ratios of the different sides of a triangle.