How do we solve trigonometric equations with more than one function?

Aim: How do we solve trigonometric equations with more than one function? 

We can solve trigonometric equations with more than one function by using trigonometric identities or reciprocal identities, so we can have one function and can easily solve by factoring. 

Example #1:
5 sin^2θ-9 cosθ-3=0

1) Replace the sine using the trig identity 1- cos^2θ  which is equivalent to sine. Doing this will allow you to work with one function.  

sin^2θ+cos^2θ=1
         - cos^2θ    -cos^2θ

    sin^2θ= 1-cos^2θ

5 (1- cos^2θ)-9 cosθ-3=0

2) Distribute the 5 into 1- cos^2θ so you can have a quadratic equation. 

5 - 5 cos^2θ-9 cosθ-3=0

3) Simplify the like term and arrange the equation in terms of ax^2+bx+c=0

-5 cos^2θ-9 cosθ+2=0
Multiply by -1 to get rid of the negative sign in front of the coefficient. 
-1(-5 cos^2θ-9 cosθ+2)=0(-1)

5 cos^2θ+9 cosθ-2=0

4)Now factor the equation by factoring them by grouping.

5 cos^2θ+9 cosθ-2=0

Multiply 5 times -2 and get -10. Than look for two numbers that multiply to -10 and add up to 9. Two number that add up to 9 and multiply to -10 are 10 and -1. Replace them with 9 cosθ.

5 cos^2θ+10 cosθ-1 cosθ-2=0

Now, group them into two pairs and factor by using common factors.

5 cos^2θ+10 cosθ                                      -1 cosθ-2
5 cosθ ( cosθ+2)                                    -1(cosθ+2)

Finally, you have (5 cosθ-1)(cosθ+2)=0

5) Solve for the solutions.

5 cosθ-1=0                                                cosθ+2=0
          +1  +1                                                      -2  -2

5/5 cosθ=1/5                                               cosθ= -2

cosθ=1/5

6)Find degrees where cosθ=1/5 and cosθ=-2

cosθ=1/5 in 78.5° and 281.5°                    cosθ= -2 has no solution
 (Do inverse cosine of 1/5 and you will get                                     because the unit circle 
78.5° and later subtract it from 360 to get                                     only goes up to 1. 
281.5°)
               
7) The solutions are 78.5° and 281.5°.

Example #2:

7 cosθ +1= 6 secθ

1)Replace secant with reciprocal of cosine.


7 cosθ +1= 6/cosθ

2) Multiply both side by cosine to get rid of the cosine from the denominator.

cosθ(7 cosθ +1)= (6/cosθ)cosθ 

7 cos^2θ +cosθ= 6

3) subtract 6 from both side so you can create a quadratic equation.

 7 cos^2θ +cosθ= 6

                      -6       -6

7 cos^2θ +cosθ-6=0

4) Now solve the rest of the problem by factoring by grouping and 

solving for the degree solutions as shown in the previous problem.