tan= 3/4 find the values of sin,cos,and tan of x/2

Question

if tan = 3/4 (in pi-3pi/2) what is the sin, tan, and cos of x/2. 

Answer 1

Hi Erin,

This a 3-4-5 reference right triangle. The angle terminates in quadrant III. The angle is 180+37 = 217 degrees. Both sinx and cosx are negative here.

sinx = -3/5
cosx = -4/5

 

To find sin, cos and tan x/2, we need to use half-angle identities. Half of 217 is 108.5 deg and is in Quadrant II so that will determine the sign. cosx and tanx are negative and sinx is positive in Quadrant II.

 

cos(x/2) = sqrt((1+cosx)/2)

sin(x/2) = sqrt((1-cosx)/2)

 

cos(x/2) = sqrt((1+(-4/5)/2)

= sqrt((1/5)/2)

= - sqrt(1/10)

 

sin(x/2) = sqrt((1-(-4/5)/2)

= sqrt((9/5)/2)

= sqrt(9/10)

 

tan(x/2) = sin(x/2) / cos (x/2)

= sqrt(9/10) / - sqrt(1/10)

= -3