If tan x=8/15, and 0°<x<90°, find cos(x/2)

Answer 1

Draw the coordinate system.

For tan (theta)=8/15 draw the following triangle:

Draw a radius in the first quadrant with end point (15,8). Label the x-coordinate 15, and the y-coordinate 8.

Compute the radius: r=sqrt(15^2+8^2)=sqrt( 289) = 17

Now you have a triangle with all sides labeled. In particular, you can find cos(theta)=15/17.

Now focus on the angle theta/2. It is still in quadrant 1. Therefore the value for cos(theta/2) is positive.

Next use the identity: cos(theta/2)=+sqrt{ ( 1+cos(theta) ) / 2 } =sqrt{ ( 1+ (15/17) ) / 2 } =0.9701