Use the sun and difference formula cosine

Question

you are given that cos(A)= 3/5 with A in quadrant 1, and sin (B)= -24/25, with B in Quadrant 3. Find cos(A-B)

Answer 1

cos(A - B)=cos(A)cos(B) + sin(A)sin(B)

cos(A)=3/5

sin(A)=(1 - (3/5)^2)^0.5 or 4/5

sin(B)=-24/25

cos(B)=(1 - (-24/25)^2)^0.5 or -7/25

cos(A+B)=3/5 × -7/25 + 4/5 × -24/25 or -117/125

Answer 2

In quadrant 1 both sin and cosine are positive.

In quadrant 3 both sine and cosine are negative.

cos(A-B)=cosAcosB+sinAsinB

sinA=√(1-cos²A)=√(1-(3/5)²)=√(1-9/25)=√(16/25)=4/5

cosB=-√(1-(24/25)²)=-√(1-576/625)=√(49/625)=-7/25

Substitute and get

cos(A-B)=-21/125-96/125=-117/125

Answer 3

Use the sum and difference formula cosine: you are given that cos(A) = 3/5 with A in quadrant 1, and sin (B) = –24/25, with B in Quadrant 3. Find cos(A – B) = ??

Detailed Solution:

Angle A in Quadrant 1 (All: sin, cos, tan, csc, sec, cot positive)

Angle B in Quadrant 3 (sin, cos, csc, sec, negative; tan,cot positive)

cos(A – B) = cos(A) cos(B) + sin(A) sin(B)

= (3/5)(–7/25) + (4/5)(–24/25)

= (–21/125) + (–96/125)

= –117/125