Question

Calculus homework, plzzzzz helplppp

Answer 1

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