SOLUTION: Find the exact value of the function tan(beta/2), given tan(beta)= -((sqrt 5)/2), with 90 degrees < beta

SOLUTION: Find the exact value of the function tan(beta/2), given tan(beta)= -((sqrt 5)/2), with 90 degrees < beta < 180 degrees

Question 821433: Find the exact value of the function
tan(beta/2), given tan(beta)= -((sqrt 5)/2), with 90 degrees < beta < 180 degrees
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

This tells us that terminates in the second quadrant, where x-coordinates are negative and y-coordinates are positive.

At this point a diagram may be helpful:

Draw an angle in standard position which terminates in the 2nd quadrant. This angle will be .
From somewhere on the terminal side draw a perpendicular down to the x-axis. This perpendicular, the x-axis and the terminal side form a right triangle.
We've been given that . Since tan is opposite/adjacent, we want to label the opposite site with the numerator of and the adjacent side with the denominator. But where does the "=" go? Answer: Since we are in the 2nd quadrant, the minus should go with the adjacent side (x-axis). So label the opposite side, the perpendicular, as and label the adjacent side (x-axis) as -2.

For we will use the tan((1/2)x) identity:

From this we can see that we will need and . For both sin and cos we need the hypotenuse. So we use the Pythagorean Theorem to find the hypotenuse:
(where "h" represents the hypotenuse)
Simplifying...

(ignoring the negative square root of 9 since hypotenuse's are never negative).

Now that we have the hypotenuse, we can find the sin, opposite/hypotenuse, and the cos, adjacent/hypotenuse, of :

And finally we can find :

Substituting in the values we found for sin and cos:

Simplifying...

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