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Question: How do you solve \(\sin (x) = 0.5\)?...

How do you solve sin⁡(x)=0.5\sin (x) = 0.5?

Explanation

Solution

In order to determine the value of the above question, convert the decimal into fraction and use the trigonometric table to find the angle which is in the interval [−π2,π2]\left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right] whose sine is12\dfrac{1}{2} to get the required result.

Complete step by step solution:
Given,
sin⁡(x)=0.5\sin (x) = 0.5
Write decimal value into the fraction
sin⁡(x)=0.5 sin⁡(x)=510 sin⁡(x)=12  \sin (x) = 0.5 \\\ \sin (x) = \dfrac{5}{{10}} \\\ \sin (x) = \dfrac{1}{2} \\\
Transposing sine from LHS to RHS
x=sin⁡−1(12)x = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right)
We know that sin⁡−1θ{\sin ^{ - 1}}\theta denotes an angle in the interval [−π2,π2]\left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right] whose sine is xx for x∈[−1,1].x \in \left[ { - 1,1} \right].
Therefore,
x=sin⁡−1(12)x = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right)= An angle in [−π2,π2]\left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right], whose sine is 12\dfrac{1}{2}.
From the trigonometric table we have,
sin⁡(π6)=12\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}
Transposing sin from left-hand side to right-hand side
x=sin⁡−1(12)=π6x = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right) = \dfrac{\pi }{6}
Therefore, the value of x is equal to π6\dfrac{\pi }{6}.

Note:
1. In Mathematics the inverse trigonometric functions (every so often additionally called anti- trigonometric functions or cyclomatic function) are the reverse elements of the mathematical functions In particular, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are utilized to get a point from any of the point's mathematical proportions. Reverse trigonometric functions are generally utilized in designing, route, material science, and calculation.
2. In inverse trigonometric function, the domain are the ranges of corresponding trigonometric functions and the range are the domain of the corresponding trigonometric function.
3. Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus.
4. Periodic Function= A function f(x)f(x) is said to be a periodic function if there exists a real number T > 0 such that f(x+T)=f(x)f(x + T) = f(x) for all x.