360 degrees <theta< 450 degrees and tan theta= 1/7. Find in simplest radical form the functional value of sec(theta)

Question

360 degrees <theta< 450 degrees and tan theta= 1/7. Find in simplest radical form the functional value of sec(theta)

Answer 1

Let x = theta

 

tan(x) = 1 / 7

 

We can use a right triangle to find sine and cosine.   We know that tan(x) is opposite side over adjacent side.  Let's use Pythagorean theorem to find the missing side, the hypotenuse.

 

 

1² + 7² = c²

1 + 49 = c²

50 = c²

√50 = c

√25 √2 = c

5√2 = c

 

Since we need sec(x), which is 1 / cos(x), we can use the value of c that was obtained.  Cosine is adjacent side over the hypotenuse.

 

1 / cos(x) = 1 / (7 / (5√2))

               = (5 / 7)√2

 

Now the angle interval is between 360 and 450.  If we subtract these angles, we obtain a difference of 90 degrees.

 

So now out interval using the smallest angle is  0 < x < 90.  This interval is in the 1st quadrant.  Since all trig functions are positive in this quadrant,

 

sec(x) = 1 / cos(x) = (5 / 7)√2