Use an identity to find the exact value of the expression. Tan(3pi/4 + 2pi/3)

Answer 1

To find the value of the expression tan(3π/4 + 2π/3), we can use the tangent addition formula:

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

In this case, A = 3π/4 and B = 2π/3, so:

tan(3π/4 + 2π/3) = (tan(3π/4) + tan(2π/3))/(1 - tan(3π/4)tan(2π/3))

Now, let's find the tangent values for each angle:

tan(3π/4) = 1
tan(2π/3) = √3

Substituting these values back into the formula, we get:

tan(3π/4 + 2π/3) = (1 + √3)/(1 - 1*√3)

Next, multiply the numerator and denominator by the conjugate of the denominator:

(1 + √3)(1 + √3)/(1 - 1*√3)(1 + √3)

Simplifying the numerator and denominator:

(1 + √3)(1 + √3) = 1 + 2√3 + 3 = 4 + 2√3
(1 - √3)(1 + √3) = 1 - (√3)^2 = 1 - 3 = -2

Finally, evaluate the expression:

tan(3π/4 + 2π/3) = (4 + 2√3)/-2 = -2 - √3

Therefore, the exact value of tan(3π/4 + 2π/3) is -2 - √3.

Answer 2

you lost track of a minus sign somewhere.
tan(3π/4 + 2π/3) is 2 + √3

Answer 3

Apologies for the error in the previous response.

To find the value of the expression tan(3π/4 + 2π/3), we can once again use the tangent addition formula:

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

In this case, A = 3π/4 and B = 2π/3, so:

tan(3π/4 + 2π/3) = (tan(3π/4) + tan(2π/3))/(1 - tan(3π/4)tan(2π/3))

Now, let's find the tangent values for each angle:

tan(3π/4) = -1
tan(2π/3) = √3

Substituting these values back into the formula, we get:

tan(3π/4 + 2π/3) = (-1 + √3)/(1 - (-1)*(√3))

Simplifying the numerator and denominator:

(-1 + √3)(1 + √3) = -1 - √3 + √3 - (√3)^2 = -1 + 2√3 - 3 = -4 + 2√3

(1 - √3)(1 + √3) = 1 - (√3)^2 = 1 - 3 = -2

Finally, evaluate the expression:

tan(3π/4 + 2π/3) = (-4 + 2√3)/(-2) = 2 - √3

Therefore, the exact value of tan(3π/4 + 2π/3) is 2 - √3.