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### Special Angles

```
Date: 01/18/2002 at 19:52:37
From: Patricia Henry
Subject: Trigonometry

I am taking a 1st year math course at university. I have been through
the text, my notes, and math software (The Princeton Review Math
Library) but I cannot figure this out. In the text, a sample problem
includes:

tan(theta) = -sqrt(3)

theta = 2pi/3

I cannot figure out how the solution for theta was found.  Can you
tell me?  I need to know in order to answer the questions at the end
of the chapter.

Thanks,
Patricia
```

```
Date: 01/18/2002 at 22:54:32
From: Doctor Peterson
Subject: Re: Trigonometry

Hi, Patricia.

This is one of the "special angles" whose trig functions should be
familiar
to you:

30-60-90 and 45-45-90 Triangles
http://mathforum.org/dr.math/problems/kristina3.15.99.html

Remembering Trig Functions
http://mathforum.org/dr.math/problems/kim.09.27.01.html

If you don't recall those offhand, but suspect (since they asked) that
the answer must be simple, you can work backward and construct the
triangle. First, ignore the signs and just make a right triangle where
the ratio of the legs is sqrt(3):

+
/|
/ |
/  |sqrt(3)
/   |
/A   |
+-----+
1

How can we see if this is a familiar triangle? Use Pythagoras to find
the hypotenuse:

1^2 + sqrt(3)^2 = 1 + 3 = 4

so the hypotenuse is 2. Hmmm ... that's twice the bottom leg, so if we
double the right triangle, we get ...

+
/|\
/ | \
2/  |  \2
/   |   \
/A   |    \
+-----+-----+
1     1

... an equilateral triangle! So angle A must be 60 degrees.

Now you have to handle the sign. To do this, picture a unit circle.
The tangent will be negative in the second and fourth quadrants,
so we have

+
|\
| \
sqrt(3)|  \
|   \
|  60\ theta
+-----o-----------
-1

So theta is 180-60 =
120 degrees; or, in radians, pi - pi/3 = 2pi/3.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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