If sec ø = 5/4 and 0º<ø<90º how do you find sec 2ø ?

1 Answer
Feb 27, 2018

sec(2theta) = 25/7 for sec(theta) = 5/4

Explanation:

sec(theta) = 5/4

Recall that sec theta = 1/cos theta. Because of this, we can simply take the reciprocal of both sides so we can work with functions we're more used to seeing.

color(blue)(cos(theta) = 4/5)

We are looking for color(green)(sec(2theta)), which can also be written in terms of trigonometric functions we are more familiar with.

color(green)(sec(2theta))

= 1/cos(2theta)

The double angle identity for cosine states that color(red)(cos(2theta) = 2cos^2(theta) - 1).

= 1/color(red)(2cos^2(theta) - 1)

Interestingly, this means that we don't actually have to solve for theta to find the value of color(green)(sec(2theta)).

= 1/(2(color(blue)cos(theta))^2 - 1)

= 1/(2(color(blue)(4/5))^2 - 1)

= 1/(32/25 - 1)

= 1/(7/25)

= 25/7

therefore sec(2theta) = 25/7 for sec(theta) = 5/4.