Exact value of a right triangle?

Question

Use the figure to find the exact value of the trigonometric function.

 

The figure is a right triangle:

Hypotenuse: √17

Adjacent: 4

Opposite: 1

 

sec(2Θ)=?

Answer 1

Hi, Taka!

 

To answer this question, you need to know which basic trig function goes with the reciprocal trig function of secant.  You will also need to have your double angle formulas handy.

 

Quick recap on basic vs. reciprocal trig functions:

 

Basic Trig Functions (SOH CAH TOA)

sinθ = opposite/hypotenuse

cosθ = adjacent/hypotenuse

tanθ = opposite/adjacent

 

Reciprocal Trig Functions (just flip all of the fractions above)

1/sinθ = cscθ = hypotenuse/opposite

1/cosθ = secθ = hypotenuse/adjacent

1/tanθ = cotθ = adjacent/opposite

 

Now, for the question you asked:

 

Secant is the reciprocal trig function associate with cosine, so...

sec(2θ) = 1/cos(2θ)

 

In order to find the value for cos(2θ), we need to know the formula.  If you look on your reference tables, you will see that cos(2θ) has three different options for which formula you use.  It won't matter which one you use, so pick the one you are most comfortable with.  Here are the options, in case you don't have this in front of you:

cos(2θ) = cos²θ - sin²θ

cos(2θ) = 2cos²θ - 1

cos(2θ) = 1 - 2sin²θ

 

I'm going to use the second option: cos(2θ) = 2cos²θ - 1.  The next step is to substitute the formula into my equation from above:

sec(2θ) = 1/cos(2θ) = 1/(2cos²θ - 1)

 

Now, cosθ = adjacent/hypotenuse = 4/√17.

 

sec(2θ) = 1/cos(2θ) = 1 / (2cos²θ - 1)

= 1 / (2*(4/√17)² - 1)

= 1 / (2*(16/17) - 1)

= 1 / (32/17 - 1) = 1 / (32/17 - 17/17)

= 1/ (15/17)

= 17/15

 

I hope this helps!

 

- Patty

Answer 2

sec 2θ = 1 / cos 2θ

sec 2θ = 1 / (cos²θ - sin²θ)

 

cos θ = 4/√17, cos²θ = 16/17

sin θ = 1/√17, sin²θ = 1/17

 

sec 2θ = 1 / (15/17)

sec 2θ = 17/15