Use the information about the angle theta 0 or equal to thet

Use the information about the angle theta, 0< or equal to theta < or equal to 2 pi, to find the exact value of each trignometric function.

tan theta= -9, sin theta < 0

(a) sin (2 theta)

(b) cos (2 theta)

(c) sin theta/2

(d) cos theta/2

Solution

given tan theta= -9, sin theta < 0,0< or equal to theta < or equal to 2 pi

from the information tan and sine are negative so theta lies in 4 th quadrant => cos theta is positive

tan theta= -9/1

tan theta= oppositeside /adjacent side =-9/1

opposite side =9 ,adjacent side =1

by pythogorus theorem hypotenuse²=opposite side²+adjacent side²

hypotenuse²=81+1

hypotenuse =82

cos theta =adjacent side /hypotenuse =1/82,sin theta =opposite side /hypotenuse =-9/82

(a) sin (2 theta)=2sintheta costheta =2*(-9/82)(1/82) =-18/82 =-9/41

(b) cos (2 theta)=2(cos²theta) -1 =2(1/82)² -1 =(1/41) -1 =-40/41

0< or equal to theta < or equal to 2 pi

0< or equal to theta/2 < or equal to pi

in 0< or equal to theta/2 < or equal to pi , sine is positive

(c) sin theta/2 =[[1-costheta]/2] =[[1-(1/82)]/2] =0.667

chnen cos theta >0 ,0< or equal to theta/2 < or equal to pi , cos is negative

(d) cos theta/2=[[1+costheta]/2] =-[[1+(1/82)]/2] =-0.745