Use reference angles to find the exact value of the following expression. Show your steps.
tan 4π/3
To find the exact value of tan 4π/3, we first need to determine its reference angle.
The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis.
In this case, the angle is 4π/3, which is greater than π (180 degrees) but less than 2π (360 degrees). It falls in the second quadrant.
To find the reference angle, we can subtract the angle from a full revolution (2π) or 360 degrees.
Reference angle = 2π - 4π/3
Reference angle = 6π/3 - 4π/3
Reference angle = 2π/3
Now that we have the reference angle, we need to determine the sign of the tangent in the second quadrant.
In the second quadrant, sine (sin) is positive, and cosine (cos) is negative.
Since tan = sin/cos, the tangent will be positive.
Now we can use the reference angle (2π/3) to find the exact value of the expression.
Using the special right triangle with angles of π/3, we know that the opposite side (y) is √3 and the adjacent side (x) is 1.
Thus, we can write:
tan(2π/3) = y/x
tan(2π/3) = √3/1
tan(2π/3) = √3
So, the exact value of the expression tan 4π/3 is √3.