were doing sum and difference identities
tan (225 degrees+60 degrees)
i got to the part where its 1+radical 3/1-radical 3 ..now what?
actually that is your answer
(1 + √3)/(1 - √3)
looking at the answer code you probably saw something different and thought you were wrong.
How about rationalizing our denominator ?
(1 + √3)/(1 - √3)
= (1 + √3)/(1 - √3)*(1+√3)/(1+√3)
= (1+√3)^2 /(1-3)
= -(1+√3)^2/2
To simplify the expression tan(225° + 60°), you can use the sum angle identity for tan.
The sum angle identity for tan is:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
Applying this identity to your expression, with A = 225° and B = 60°:
tan(225° + 60°) = (tan 225° + tan 60°) / (1 - tan 225° * tan 60°)
Now, you need to find the tangent values for 225° and 60°.
You can use the unit circle to find the tangent values. Recall that the unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0). The tangent value for an angle is given by the y-coordinate divided by the x-coordinate of the corresponding point on the unit circle.
For 225°, we need to find the point on the unit circle. To do this, we can take the 45° rotation in the clockwise direction from the 180° point on the unit circle. This corresponds to the point (-√2/2, -√2/2) on the unit circle.
Therefore, tan 225° = (-√2/2) / (-√2/2) = 1.
For 60°, we can find the point on the unit circle by taking the 60° rotation in the counterclockwise direction from the positive x-axis. This corresponds to the point (√3/2, 1/2) on the unit circle.
Therefore, tan 60° = (1/2) / (√3/2) = 1/√3 = √3/3.
Substituting these values back into the expression:
tan(225° + 60°) = (1 + √3/3) / (1 - 1 * √3/3)
Simplifying the denominator:
1 - 1 * √3/3 = 1 - √3/3.
Now, multiplying the numerator and denominator by the conjugate of the denominator:
(1 + √3/3) / (1 - √3/3) * (3/3) = (3 + √3) / (3 - √3).
Therefore, tan(225° + 60°) simplifies to (3 + √3) / (3 - √3).