Answer:
(4 + 3√3) / 10
(-3 − 4√3) / 10
(48 + 25√3) / 39
Step-by-step explanation:
First we need to find sin α and cos α.
One way is to recognize that tan α = -4/3 corresponds to a 3-4-5 triangle. Since α is in the second quadrant:
sin α = 4/5
cos α = -3/5
Alternatively, we can use Pythagorean identities:
1 + tan² α = sec² α
1 + (-4/3)² = sec² α
sec α = -5/3
cos α = -3/5
Then use definition of tangent to find sine:
tan α = sin α / cos α
-4/3 = sin α / (-3/5)
sin α = 4/5
Next, we need to use the same process to find sin β and tan β.
Since cos β = 1/2 and β is in the fourth quadrant, β = 5π/3. So sin β = -√3/2, and tan β = -√3.
Or, using Pythagorean identities:
sin² β + cos² β = 1
sin² β + (1/2)² = 1
sin β = -√3/2
Using definition of tangent:
tan β = sin β / cos β
tan β = (-√3/2) / (1/2)
tan β = -√3
Now we're ready to start solving using angle sum/difference formulas.
4. sin(α+β)
sin α cos β + sin β cos α
(4/5) (1/2) + (-√3/2) (-3/5)
4/10 + 3√3/10
(4 + 3√3) / 10
5. cos(α−β)
cos α cos β + sin α sin β
(-3/5) (1/2) + (4/5) (-√3/2)
-3/10 − 4√3/10
(-3 − 4√3) / 10
6. tan(α+β)
(tan α + tan β) / (1 − tan α tan β)
(-4/3 + -√3) / (1 − (-4/3) (-√3))
(-4/3 − √3) / (1 − 4√3/3)
(-4 − 3√3) / (3 − 4√3)
Rationalizing the denominator:
(-4 − 3√3) / (3 − 4√3) × (3 + 4√3) / (3 + 4√3)
(-12 − 16√3 − 9√3 − 36) / (9 − 48)
(-48 − 25√3) / -39
(48 + 25√3) / 39
