When asked for "exact" values, that means don't give the answer in terms of a decimal that can be found on your calculator. We can use either the 30-60-90 triangle or the 45-45-90 triangle to find the exact values. The Pythagorean triple for a 30-60-90 is (1, sqrt(3),2) and the Pythagorean triple for the 45-45-90 is (1,1,sqrt(2). So let's first convert our angle to degrees. Use the fact that there are 180 degrees in pi. [tex] \frac{4\pi}{3}*\frac{180}{\pi} [/tex]. The pi cancels leaving us in terms of degrees, which is what we want. Simplify that to find that degree equivalent is 240. The first quadrant measures from 0 to 90 degrees, the second quadrant measures from 90 to 180 degrees, the third quadrant measures from 180 to 270, and the fourth quadrant measures from 270 to 360. Our angle measure of 240 lies in the third quadrant. From the 180 degree line we go 60 degrees into the third quadrant. So the reference angle we are working with is a 60 degree angle. The side across from the 60 is sqrt(3). the side across from the 30 is 1, and the side across from the right angle, which is the hypotenuse, is 2. The sin of an angle relates the side opposite the angle to hypotenuse in a ratio. Keep in mind that the x values in this quadrant are negative as are the y values. The hypotenuse is NEVER negative. That means the sin of that angle is [tex] -\frac{\sqrt{3}}{2} [/tex]. The cosine of the angle will relate the side adjacent to the hypotenuse in a ratio. That means that the cosine of the angle is [tex] -\frac{1}{2} [/tex]. The tangent relates the side opposite to the side adjacent in a ratio. So the tangent of the angle is [tex] \frac{-\sqrt{3}}{-1} [/tex]. Since negative divided by negative is positive, and anything divided by 1 is equal to itself, the tangent of the angle is [tex] \sqrt{3} [/tex]. There you go!
