trigonometry solutions for equations

Question

I need to find solutions for trigonometric equations for college algebra and trig and my prof did not give any notes on how to solve this type of equation there are 5 equations and I don't know how to solve any of them without the notes, I'm not asking for the exact answer just a way to solve them.

cos(0) = - 1/2

2sin(0)+1 = 0

4sin²(0)-3 = 0

4cos²(0)+cos(0)-3 = 0

sin(0)tan(0)+sin(0) = 0

Answer 1

The key to understanding how to solve these is that you need to memorize and understand the Unit Circle. In the Unit Circle there are key (major) angles for which the exact values of trig ratios are given. For instance, you learn that the cosine of 120° or 2π/3 radians is -1/2 and that the cosine of 240° or 4π/3 radians is also -1/2. This allows you to solve the first equation. Note that you may also use inverse function relationships and your calculator as follows (with some limitations):

cos(θ) = -1/2

cos^-1(cos(θ)) = cos^-1(-1/2)

θ = 120° however, without understanding the Unit Circle, you wouldn't understand that there is a 2nd answer of 240°

Take some time to study your textbook and some videos regarding the Unit Circle. Khan Academy has some good resources.

Answer 2

First step is that you have to isolate the term with the variable onto one side of the equation

For problem 1, the cos is already isolated. Use the unit circle to find the angles where the cos is -1/2.

For problem 2, you have to isolate sin 0 as your first step. Subtract 1 from both sides of the equation, divide by 2. Then take that value and find the angles that give you that sin value

For problem 3, isolate the sin again. However this time, since it is squared, you have to take the square root of both sides of the equation. When you take the square root, that will give you +/- values. So you will have to find the angles on the unit circle that give you both positive and negative answers.

For problems 4 and 5, these are more complicated. You have to factor these, set the factors equal to zero and solve as you have solves the other ones previous.