Pre-cal: Find the exact value of each of the following: a) sin (A+B) b) cos(A+B) c) sin(A-B) d) tan (A-B)

Question

Find the exact value of each of the following:

a) sin (A+B)

b) cos(A+B)

c) sin(A-B)

d) tan (A-B)

under the given conditions:

i sin A = 3/5 , 0<A<pi/2

cos B = (2sqrt(3))/5 , -pi<B<0

ii tan A = 5/12 , pi<A< 3pi/2

sin B = -1/2 , pi<B<3pi/2

Answer 1

Calculating the exact value of sin(A + B), cos(A + B), sin(A - B) and tan(A - B) requires the use of the trig angle subtraction identities:

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

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

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

tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

and since we are only given sin(A) and cos(B), we must calculate the others from this information.

Given that sin(A) = 3/5 and that the angle A is in Q1 and knowing that sin(θ) = opp/hyp, we can draw this sketch:

And we can solve for x using the Pythagorean Theorem:

x = √(5² - 3²) = √(25 - 9) = √16 = 4

That means:

cos(A) = 4/5

tan(A) = 3/4

Then, given that cos(B) = 2√3/5, and angle B is in either Q3 or Q4, and knowing that cosine is positive in Q4 but negative in Q3 and also knowing that cos(θ) = adj/hyp we can draw a similar sketch of angle B:

And we can solve for y using the Pythagorean Theorem:

y = √(5² - (2√3)²) = √(25 - 4•3) = √13 however this is going downwards so y = -√13

That means:

sin(B) = -√13/5

tan(B) = -√13/(2√3) = -√39/6

We can now plug these values into the identities:

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

sin(A + B) = (3/5)(2√3/5) + (4/5)(-√13/5)

sin(A + B) = (6√3/25) + (-4√13/25)

sin(A + B) = (6√3 - 4√13)/25

You do the others just the same. Use the values we got to substitute into the identities and simplify.