Determine the point(s) of intersection of f(theta) = 8sec(2theta) + 4, and g(theta) = 5sec(2theta) - 11 that lie on the interval 0 - 360 degree.

Question

It was on my grade 11 math test, a thinking question, and our teacher never gave answers for it. I'm worried it will be a question for the exam. I'm not sure how to go about it, other than making one side equal the other. 

Answer 1

8sec(2θ)+4 = 5sec(2θ)-11

 

3sec(2θ) = -15

 

sec(2θ) = -5

 

cos(2θ) = -0.2

 

Let x = 2θ  Then cosx = -0.2

 

cos^-1(0.2) = 78.46°

 

Since cosx < 0, x lies in QII or QIII

 

x = 180° - 78.46° = 101.54° or 180° + 78.46° = 258.46°

 

Since x = 2θ, we have:

 

2θ = 101.54° + k360° or 258.46° + k360°, k = 0, ±1, ±2, ...

 

So, θ = 50.77° + k180° or 129.23° + k180°

 

If k = 0, we get θ = 50.77°, 129.23°

If k = 1, we get θ = 230.77°, 309.33°

 

For k > 1, the values of θ are greater than 360° and for k < 0 the values of θ are negative, so we need not consider those values.  

Answer 2

Let θ = x

 

 

f(x) = 8sec(2x) + 4

g(x) = 5sec(2x) - 11

 

In order to find points of intersection, we set f(x) = g(x).

 

8sec(2x) + 4 = 5sec(2x) - 11

 

Move all the terms to the left side of equation to make the right side equal to zero. Subtract 5sec(2x) , and add 11 on both sides of equation.

 

3sec(2x) + 15 = 0

 

sec(2x) is the same as 1/cos(2x).  This is also a double angle.

 

1 / cos(2x) = 1 / (cos²(x) - sin²(x)) = 1 / (cos²(x) - (1 - cos²(x))) = 1 / (2cos²(x) - 1)

 

 

(3 / (2cos²(x) - 1)) + 15 = 0

 

 

The least common denominator of the left side of equation is   2cos²(x) - 1

 

(3 + 30cos²(x) - 1) / (2cos²(x) - 1) = 0

 

(2 + 30cos²(x)) / (2cos²(x) - 1) = 0

 

 

Set the numerator equal to zero to find the solutions.

 

2 + 30cos²(x) = 0                             

 

30cos²(x) = -2                                                     

                                                                          

    cos²(x) = -1/15                                                   

 

     cos(x) = -1/√(15)                                               

 

     cos(x) = -√(15) / 15                                            

          

           x = 75.04

 

cosine is negative in the second and third quadrant.  Let's accept the following values of x.

 

x = 180 - 75.04 = 104.96°

x = 180 + 75.04 = 255.04°