Mathematics

Trigonometry Review

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SOHCAHTOA

For any right triangle (i.e., a triangle with a $90^\circ$ angle between two of its segments), we can apply the following equations to determine the length of segments that comprise the triangle or the degree of the acute angles in the triangle.

 

sohcahtoa

SOH: \begin{equation*} \sin(\theta) = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}} \end{equation}

CAH: \begin{equation*} \cos(\theta) = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}  \end{equation}

TOA: \begin{equation*} \tan(\theta) = \frac{\mathrm{opposite}}{\mathrm{adjacent}}  \end{equation}

Equation for a Circle

A circle is defined as the set of points lying at a fixed distance from some center point. If we set the center of a circle with a radius $r$ to be the origin of a 2D Cartesian coordiante system, then we can use the Pythagorean theorem to find any set of points $P:(r_x, r_y)$ that lie along the perimeter of our circle. Equation of a circle

If we want to know the $r_x$ and $r_y$ coordinates associated with some $0 \leq \theta \leq 2\pi$ radians, we can use our first two SOHCAHTOA equations:

  • $r_x= r \cos(\theta)$
  • $r_y = r \sin(\theta)$

Aside: to convert between radians and degrees, use the equality $\pi \, \mathrm{rad} = 180^\circ$

  • Thus, $1 \, \mathrm{rad} = (\frac{180}{\pi})^\circ \approx 57.2958^\circ$

    radian
    $1 \, \mathrm{rad} \approx 57.3^\circ$

The Unit Circle

Unit Circle

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