Trigonometry Handout

Trigonometry
π
Note: You could have also formed the right triangle with θ = and made the appropriate
4
sign changes for the quadrant you were in.
Example 4: Let sin − 5
3π
θ = where π < θ < . Find the values of the other trigonometric
6
2
functions.
Solution:
5
6
x =
11
θ
We know that θ is in the 3 rd Quadrant.
Thus, only tangent and cotangent have
positive signs. Every thing else will be
negative. For now we will just drop the
signs to form a right triangle with hyp = 6
and opp = 5 since sine is -5/6
By the Pythagorean Theorem, we know that
2 2 2 2
2
x + 5 = 6 ⇒ x + 25 = 36 ⇒ x = 11 ⇒ x =
11
So, we get the following ratios for the other 5 trig functions
11 5
6
6
cos θ = − tanθ
= cscθ
= − secθ
= − cotθ
=
6
11 5
11
11
5
The Unit Circle:
The unit circle is a circle centered at the origin with radius 1, hence the name unit
circle. It allows us to easily find the values of sine and cosine, and thus the rest of the
trigonometric functions quickly and easily. There is a picture of the unit circle on the
following page (Figure 11). The x-coordinates are the values of cosine and the y-
coordinates are the values of sine.
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