# Trigonometric Functions of Any Angle

Math Analysis - Unit Circle - How to create the Unit Circle from scratch

Tip: This isn't the place to ask a question because the teacher can't reply.

1 of 2 videos by AJ Speller

## Key Questions

• 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°).
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)

• The reference angle is the smallest angle you can make between the x-axis and the terminal side of an angle. You always look at the x-axis as your frame of reference.

For example, for an angle measuring $x$ degrees,

if $x < 90$, the terminal side is in the first quadrant, the reference angle is the acute angle formed counterclockwise from the x-axis to that terminal side.

If $90 < x < 180$, the terminal side is in the second quadrant, the reference angle is equal to $180 - x$.

If $180 < x < 270$, the terminal side is in the third quadrant, the reference angle is equal to $x - 180$.

If $270 < x < 360$, the terminal side is in the fourth quadrant, the reference angle is equal to $360 - x$.

• The good thing about a unit circle is the fact that any point on it already has the coordinates $\left(\cos x , \sin x\right)$. This comes from the fact that $x = r \cos$ and $y = r \sin x$, and that $r = 1$ for the unit circle. From this we can easily get the values for $\tan x$, $\cot x$, $\sec x$, and $\csc x$.

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