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Question: How do you find the solution to \( \csc \theta - 1 = 3\csc \theta - 11 \) if \( 0 \leqslant \theta <...

How do you find the solution to csc⁡θ−1=3csc⁡θ−11\csc \theta - 1 = 3\csc \theta - 11 if 0⩽θ<2π?0 \leqslant \theta < 2\pi ?

Explanation

Solution

Hint : As we know that the above question is related to trigonometry as sine, cosine, cosecant are the trigonometric ratios. To solve the given trigonometric equation we should know all the basic relations between the trigonometric ratios and their formulas. In this question we will bring the similar terms to the same side of the equation and then solve it.

Complete step by step solution:
As per the given question we have
csc⁡θ−1=3csc⁡θ−11\csc \theta - 1 = 3\csc \theta - 11 .
We will bring the cosine terms together in the left hand side of the equation and transfer the constant to the right hand side of the equation:
3csc⁡θ−csc⁡θ=11−1 ⇒2csc⁡θ=103\csc \theta - \csc \theta = 11 - 1 \\\ \Rightarrow 2\csc \theta = 10 .
On further solving we have csc⁡θ=102=5\csc \theta = \dfrac{{10}}{2} = 5 .
We know that the cosecant is the inverse of sine, so we have
1sin⁡θ=5 ⇒sin⁡θ=15\dfrac{1}{{\sin \theta }} = 5\\\ \Rightarrow \sin \theta = \dfrac{1}{5} .
Hence the value is sin⁡θ=15\sin \theta = \dfrac{1}{5} .
So, the correct answer is “ sin⁡θ=15\sin \theta = \dfrac{1}{5} ”.

Note : Before solving this kind of question we should have the clear concept of trigonometric ratios, identities and their formulas. We should note that in the above solution for further solving the value of theta we can convert it into the radian value which can be written as θ≈0.20136,2.9402\theta \approx 0.20136,2.9402 radians.