Exact Solution for 5sinθ + 2√3 = √3 + 3sinθ - Step by Step Guide

[aisha]

The question says find exact solution algebraically of 5sintheta+2root3=root3+3sin theta

I tried combining like terms 3sintheta and 5 sin theta and got 2sintheta on the left but don't know about the right side, and then what do I do next to solve this? HELP

 

[Andrew Mason]

The question says find exact solution algebraically of 5sintheta+2root3=root3+3sin theta

I tried combining like terms 3sintheta and 5 sin theta and got 2sintheta on the left but don't know about the right side, and then what do I do next to solve this?

It is not as hard as you think. Just use:

$$5x + 2y = y + 3x$$

and then substitute $x=sin\theta$ and $y = \sqrt{3}$

AM

 

[wais]

To solve this equation, we can follow these steps:

Step 1: Combine like terms on both sides of the equation. On the left side, we have 5sinθ and on the right side, we have √3 + 3sinθ. Combining these terms, we get 5sinθ + 3sinθ = 8sinθ and √3 + √3 = 2√3. So our equation becomes 8sinθ + 2√3 = 2√3.

Step 2: Subtract 2√3 from both sides to isolate the term with sinθ. This gives us 8sinθ = 0.

Step 3: Divide both sides by 8 to get sinθ by itself. This gives us sinθ = 0.

Step 4: Now, we need to find the values of θ that satisfy this equation. We know that sinθ = 0 when θ = 0 and θ = π. However, we also need to consider all the angles that have a reference angle of 0 or π, such as 2π, 3π, etc. So our solutions are θ = 0, π, 2π, 3π, etc.

Therefore, the exact solution to the equation 5sinθ + 2√3 = √3 + 3sinθ is θ = 0, π, 2π, 3π, etc.