Given tan A and sin B 35 A is in Quadrant 3 and B is in Q

Given tan A = and sin B = - 3/5 A is in Quadrant 3 and B is in Quadrant 4. Find a) sin 2A b) cos 2B c) tan (A+B) d) tan 2A Verify the identity sin (pi/6 + x) + sin (pi/6 - x) = cos x Write 2 cos 85 degree sin 140 degree as a sum or difference of trigonometric functions Verify the identity sin^3 = sin - cos^2 sin Verify the identity cos(x + Y) + cos(x - Y)/sin (x + Y) + sin(x - Y) = cotx

Solution

given tan A= 3/4 , sinB = -3/5

A is in 3quadrant and B is in 4quadrant

we know that sec^2(x) -tan^2(x) =1

so sec^2(A) = 1 + tan^2(A)

sec^2(A) = 1+(3/4)^2

sec^2(A) = 1 +9/16

sec^2(A) = 25/16

secA = -5/4 (since A is in 3quadrant)

now we can find cosA = 1/secA = -4/5

sin^2(A) +cos^2(A) =1

sin^2(A) = 1- cos^2(A)

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

sin^(A) = 1 - 16/25

sinA =-3/5

in the similer way we find cosB , tanB

sin^2(B) +cos^2(B) =1

cos^2(B) = 1-sin^2(B)

cos^2(B) = 1-(-3/5)^2

cosB= 4/5 (since B is in 4quadrant)

now we can find tanB = sinB/cosB

tanB= (-3/5)/(4/5)

tan(B) = -3/4

a).sin2A= 2sinA.cosA

= 2*(-3/5).(-4/5)

= 24/25

b).cos2B= cos^2(B) -sin^2(B)

= (4/5)^2 -(-3/5)^2

=16/25 -9/25

= 7/25

c). tan(A+B) = (tanA +tanB) /(1 - tanA.tanB)

=(3/4 -3/4)/(1- 3/4.-3/4)

=0

d) tan2A = (2tanA)/(1-tan^2A)

= (2*3/4) /(1- 9/16)

= (6/4)/(7/16)

= 24/7

post other questions separately, thanks