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Trigonometric Functions

April 29th, 2009 |

Author: Merrin

Trigonometry is the study of the relationships between the sides and angles of triangles. Trigonometry is something most people are familiar with before they attempt to learn calculus. Before we roll out the triangles we should review what an angle means. An angle is the measure of rotation between two rays originating from a common origin. Angles can be measured in degrees or radians. If the rays are antiparallel then by definition this is 180 degrees or $latex \pi&s=1$ radians. If the two rays form a sharp corner of half this angle which is $latex \pi/2&s=1$ radians or 90 degrees the angle is also said to be a right angle.

Conversion between angles and radians can be done with the following formula.

$\cfrac{\text{ 180 degrees}}{\pi} = 1$

In common usage, degrees are most likely used. For example if Tony Hawk just did a 900 on his skateboard. That means he did two loops of 360 which is 720 and then he went 180 more. Tony Hawk was the first person to do a 900 in competition. You could also say Tony Hawk did $latex 5\pi&s=1$, but I don’t think anyone has ever said this. The radian units are more natural for mathematics because they use the ratio of the diameter to the perimeter of the circle. The number 360 isn’t really related to anything, but it is a good numbers with lots of factors that can be divided many ways. Learn how to think interchangeably between the two units.

Example 1. Convert 45 degrees into radians

Solution 1.

$(45 \,\,\text{deg})\cfrac{\pi}{180 \,\,\text{deg}} = \cfrac{\pi}{4}$

Example 2. Convert $\cfrac{3\pi}{2}$ into degrees.

Solution 2.

$\cfrac{3\pi}{2} \cfrac{180 \,\, \text{deg}}{\pi} = 270\,\,\text{deg}$

Now let’s focus on the angles that are less that 90 degrees for a triangle with a right angle inscribed in a circle. Such a triangle is called a reference triangle, because it is an aid in calculating the trigonometric functions. If two of the sides of the triangle are known then the third can be found from the Pythagorean theorem.

If the hypotenuse is of length one then the calculation is simplified. The opposite side length is just $latex \overline{BC} = \sin \theta&s=1$ and the adjacent side length is just $latex \overline{AB}=\cos \theta&s=1$.

\begin{figure}[h]

\begin{center}

\includegraphics[width= 3in]{../images/circtrigref.pdf}

\caption{A trigonometric reference triangle inscribed in the unit circle}

\end{center}

\end{figure}

The trigonometric functions are defined from the reference triangle.

$\begin{array}{ll}\sin \theta =\cfrac{\overline{BC}}{\overline{AC}} = \cfrac{ \text{opposite}} { \text{hypotenuse} } &\cos \theta =\cfrac{\overline{AB}}{\overline{AC}} = \cfrac{ \text{adjacent}} { \text{hypotenuse} } \\\\\tan \theta =\cfrac{\overline{BC}}{\overline{AB}} = \cfrac{ \text{opposite}} {\text{adjacent} } &\csc \theta =\cfrac{\overline{AC}}{\overline{BC}} = \cfrac{ \text{hypotenuse}}{\text{opposite} } \\\\\sec \theta =\cfrac{\overline{AC}}{\overline{AB}} = \cfrac{ \text{hypotenuse}}{\text{adjacent} } &\cot \theta =\cfrac{\overline{AB}}{\overline{BC}} = \cfrac{ \text{adjacent}} {\text{opposite} }\end{array}$

The first three relations can be remembered by the pneumonic “soh-cah-toa.” SOH means Sine is Opposite over Hypotenuse and so on for CAH and TOA. Cosecant, secant, and cotangent are the reciprocals or sine, cosine, and tangent respectively.

Angles betwee 0-90 degrees or 0 to $latex \pi/2&s=1$ are said to be in the first quandrant. Angles between 90-180 degrees or $latex \pi/2&s=1$ to $latex \pi&s=1$ are said to be in the second quadrant, and so on for the next two quandrants. When $latex 2\pi &s=1$ is reached then the same pattern repeats for $latex \sin x&s=1$ and $latex \cos x&s=1$. The functions are said to be periodic with a period of $latex 2\pi&s=1$.

Some common values of the trigonometric functions at certain angles should be memorized. If you know the values for 0, $latex \pi /6&s=1$,$latex \pi /4&s=1$,$latex \pi /3&s=1$, and $latex \pi /2&s=1$ for $latex \sin x&s=1$ and $latex \cos x&s=1$ then you can do quite a bit of computations. These values can be used to find the values of the trigonometric functions between 0 and $latex 2\pi&s=1$ using symmetry.

For example, $latex \sin (\pi /6) = \sin (5 \pi /6)&s=1$. This is because the height of the inscribed reference triangle for each length is the same. As an exercise you should write out all the possible angles that can be derived from the initial set I have given you.

\begin{array}{|c| c | c |c |

$f(x)$ \rule{0in}{0.2in} & 0 & ${\pi}/{6}$ & ${\pi}/{4}$ & ${\pi}/{3}$ & ${\pi}/{2}$ \\ \hline

$\sin x\s$ & 0 & ${1}/{2} $ & ${\sqrt{2}}/{2}$ & ${\sqrt{3}}/{2}$& 1 \\

$\cos x\s$ & 1 & ${\sqrt{3}}/{2} $ & ${\sqrt{2}}/{2}$ & $1/2$ & 0 \\

$\tan x\s$ & 0 & ${\sqrt{3}}/{3} $ & $1$ & $\sqrt{3}$ & DNE \\ \hline

\end{tabular}

\caption{ Some common trigonometric values in the first quadrant }

\end{table}

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