Trigonometric Equations :)

#LlarcGrad2015

Hello everybody so I'm going to explain how to solve trigonometric equations!

When you're solving trig equations you're basically looking for the values of an unknown angle. The domain will always be 0°≤θ≤360° or 0≤θ≤2π unless otherwise stated. Ultimately we're always going to be solving for θ(theta).

In order to solve trigonometric equations you need to know the CAST rule, the unit circle, the special triangles, SOH CAH TOA, and the quadrantal angles. You need to put all this together to be able to solve trig equations. 

Unit circle

CAST rule

The steps to solve for θ:

1. isolate the trig ratio
2. find the reference angle
3. determine in which quadrant(s) the angle lies
4. check the interval given
5. state final answer(s) for θ in degrees or radians.

Okay so now we're going to solve an example question by following the 5 steps! 


Solve: 2sinθ+ √3=0 , π/2≤θ≤ 3π/2. 

Setp 1: isolate the trig ratio

Okay so the first step to solving this equation is to isolate sinθ. To do this we subtract √3 from both sides of the equation.

2sinθ+ √3=0 

2sinθ+√3-√3 =0-√3

        

             2sinθ=-√3

Now we perform the opposite operation and divide both sides of the equation by 2.

2sinθ+ √3=0 

2sinθ+√3-√3 =0-√3

        

            (2sinθ)÷2 =(-√3)÷2

                    sinθ=-√3/2

Step 2: find the reference angle

To find the reference angle first we determine where sin=√3/2 on the unit circle.  

The coordinates of a unit circle tell us that x=cos and y=sin. 

By looking at the unit circle, we can tell that sin=√3/2 in quadrants 1 and 2. 

Therefore, our reference angle will be 

located in those quadrants. 

In this case, our reference angle is π/3. 

Step 3: determine in which quadrant(s) the angle lies

To determine in which quadrant the angle lies, we look at the sign of sinθ. Whether it's positive or negative indicates where our angle lies. 

sinθ=-√3/2  Here, sine theta equals negative root three over two. Using the CAST rule,
                          we are able to determine that sine is negative in quadrants three and four. 
                          Therefore, our angles will lie in those quadrants.

Now we solve for theta! 

Q III     θ= π + π/3                                                        Q IV     θ= 2π - π/3

              θ= π/1 + π/3                                                                   θ= 2π/1 - π/3

              θ=(π/1) x 3 + π/3                                                           θ=(2π/1) x 3 - π/3

              θ=3π+π/3                                                                        θ= 6π-π/3

              θ=4π/3                                                                             θ= 5π/3

So we have θ= 4π/3, 5π/3. Next we need to check the interval so we end up with one answer 

for θ. 

Step 4: check the interval given

In our question 2sinθ+ √3=0 , π/2≤θ≤ 3π/2, the interval we're provided with is 

π/2≤θ≤ 3π/2. Our final answer has to satisfy the domain. 

                      θ= 4π/3 ✓             θ= 5π/3 X

θ= 5π/3 is rejected because of the restriction on the domain. 

Step 5. state final answer(s) for θ in degrees or radians

θ= 4π/3

Voila! That's how you solve a trigonometric equation! :)