# Trigonometric Functions of Any Angle

## Key Questions

• Well, the reference angle is the angle [the one which is the smallest] between the $x$-axis and the terminal side.

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As below:

#### Explanation:

Ordered pairs on an unit circle

Exact values of the ordered pairs of the unit circle is represented in pictorial form above in degrees and radians.

Hope this helps.

• Well, I guess you could use a special representation of the function through a sum of terms, also known as Taylor Series.
It is, basically, what happens in your pocket calculator when you evaluate, for example, #sin(30°)#.
Your calculator does this:
#sin(theta)=theta-theta^3/(3!)+theta^5/(5!)-...#
where $\theta$ must be in RADIANS.
In theory you should add infinite terms but, depending upon the accuracy required, you can normally stop at three terms.
In our case we have: $\theta = \frac{\pi}{6} = \frac{3.14}{6} = 0.523$ and:
$\sin \left(\frac{\pi}{6}\right) = \sin \left(0.523\right) = 0.523 - 0.024 + 3.26 \cdot {10}^{- 4} - \ldots = 0.499 \approx 0.5$

You can find the Taylor series for the other trigonometric functions such as:

(Picture source: www.efunda.com)