Find secant of theta if tangent of theta is the square root of 29 over 4?

If you use the pythagorean identities, of tangent ^2 +1 = secant ^2, how do you solve?

1 Answer
Apr 7, 2018

sec(theta) = (3sqrt(5))/4sec(θ)=354

Explanation:

So, what we want to solve is this:

If tan(theta) = sqrt(29)/4tan(θ)=294 ,what is sec(theta)sec(θ)?

There's a number of ways to solve this problem, but let's solve it using a trigonometric identity:

tan^2(theta) + 1 = sec^2(theta)tan2(θ)+1=sec2(θ)

If we solve this expression for sec(theta):sec(θ):

sec(theta) = sqrt(tan^2(theta) + 1)sec(θ)=tan2(θ)+1

Now, all we do is plug in the value we were given for tan(theta)tan(θ):

sec(theta) = sqrt((sqrt(29)/4)^2 + 1) = sqrt(29/16 + 1) = sqrt(45/16)sec(θ)= (294)2+1=2916+1=4516

..and that's your answer! However, we can polish this up a little so it looks nicer. For one, we can remove the radical in the denominator, since 16 is a perfect square:

=> sec(theta) = sqrt(45)/4sec(θ)=454

Now, we also know that 45 = 9*545=95. Notice that 9 is a perfect square, so we can further simplify:

=> sec(theta) = (sqrt(9) * sqrt(5))/4 = (3sqrt(5))/4sec(θ)=954=354

And that is pretty much as far as we can simplify. So, this would be your final answer.

Hope that helped :)