Need help with my webassign homework

Question

 Find the values of the trigonometric functions of θ from the information given.
sec θ = 6, sin θ < 0

Answer 1

Since secθ = 6, cosθ = 1/secθ = 1/6

 

cos²θ + sin²θ = 1, so sin²θ = 1 - cos²θ

 

Therefore, sinθ = ±√(1 - cos²θ)

 

But,  sinθ < 0.  So sinθ = -√(1 - cos²θ) = -√(1-(1/36)) = -√(35/36)

                                                                                                = -√(35)/6

 

tanθ = sinθ/cosθ = -√35

 

cotθ = 1/tanθ = -1/√35 = -√(35)/35

 

cscθ = 1/sinθ = -6/√35 = (-6√(35))/35

Answer 2

Let x = θ

 

 

sec(x) can be rewritten as  1 / cos(x).

 

1 / cos(x) = 6

 

cos(x) = 1/6

 

x = cos^-1(1/6)

x = 80.41

 

 

Now we use the condition  sin(x) < 0.   sine is negative in the 3rd and 4th quadrants. 

 

We also know that cosine is positive in the 1st and 4th quadrants. 

 

We need to find a value of x that lies in the 4th quadrant, since the equation and condition have that quadrant in common.

 

x = 360 - 80.41

x = 279.59

 

The solution is

 

x = 279.58 degress