3csc(2X)-5=0 What are the solutions for X and how did you find them out?

3 csc(2x) = 5

csc(2x) = 5/3
sin(2x) = 3/5
2x = 36.9° or 143.1°
x = 18.4° or 71.6°

Thanks Reiny. If a question similar to this one had 4 solutions, how would you know that it had 4 answers?

I cannot give a general answer to your question without seeing the actual problem

In solving trig equations, you have to consider that each trig ratio is positive in 2 different quadrants and is negative in 2 different quadrants

Usually answers are neede for the domain between 0 and 360, so the period of the trig function also has to be considered

e.g. sin (4x) = .5 would have 4 different answers between 0 and 360

for the example you gave i can get 37.5 deg and 7.5deg. what are the other solutions?

To find the solutions for X in the given equation 3csc(2X) - 5 = 0, we first need to isolate the term involving X on one side of the equation.

Adding 5 to both sides of the equation, we have: 3csc(2X) = 5.

Next, we can divide both sides of the equation by 3 to get: csc(2X) = 5/3.

To solve for X, we need to find the values of 2X for which the csc(2X) equals 5/3.

The csc(θ) function is the reciprocal of the sin(θ) function. So, we can rewrite the equation as: sin(2X) = 3/5.

Now, we can take the arcsin (or sin^(-1)) of both sides to find the values of 2X: arcsin(sin(2X)) = arcsin(3/5).

Since arcsin and sin are inverse functions, they cancel each other out, resulting in: 2X = arcsin(3/5).

Now, we need to solve for X. Divide both sides of the equation by 2: X = (1/2) * arcsin(3/5).

We can use a scientific calculator or an online tool to evaluate the arcsin(3/5) and multiply it by 1/2 to obtain the value of X.

Note: Since the sin function is periodic, there may be multiple solutions for X in a given range. To find all the solutions, we can add or subtract multiples of 180° or π to the value of X calculated above.