If tan 𝛼 = − 20 21 and cot 𝛽 = 35 12 for a second-quadrant angle 𝛼 and a third-quadrant angle 𝛽, find the following. (a) sin(𝛼 + 𝛽) (b) cos(𝛼 + 𝛽)

Question

(a)    sin(𝛼 + 𝛽)

(b)    cos(𝛼 + 𝛽)

 

Answer 1

tanA=-20/21 A in quadrant 2 sinA/cosA =-20/21, sinA =20/29, cosA=-21/29 since 20^2+21^2=29^2

cotB = 35/12 B in quadrant 3 cosB/sinB = 35/12 cosB=-35/37, sinB = -12/37 since 35^2+12^2 =37^2

sin(A+B) = sinAcosB + cosAsinB = (20/29)(35/37) + (-21/29)(12/37) =(700-252)/1073 =-448/1073=-.42

cos(A+B) = cosAcosB - sinAsinB = (-21/29)(-35/37) - (20/29)(-12/37)= (735+260)/1073 =995/1073=.91

check the answers:

A= about 180-43.6 = 136.4 degrees in quadrant 2

B= about 180+18.9 = 198.9 degrees in quadrant 3

A+B = about 335.3

sin335.3 = about -.42

cos335.3 =about .91

Answer 2

You will have to find the sine and cosine of the angles first:

tan α = -20 / 21 = y/x

In second quadrant x < 0, y >0 so we can assign x = -21, y = 20.

r = root(x² + y²) = 29

sin α = y/r = 20/29

cos α = x/r = -21/29

cot β = 35 / 12 = x/y

In third quadrant x < 0, y < 0 so we can assign x = -35, y = -12.

r = root(x² + y²) = 37

sin β = y/r = -12/37

cos β = x/r = -35/37

(a) sin(α+β) = sinα cosβ + cosα sinβ

Plug in the values and evaluate.

(b) cos(α+β) = cosα cosβ - sinα sinβ

Plug in the values and evaluate.