Section B.2 Right Triangle Trigonometry
A right triangle is a triangle containing a 90° angle.
The side opposite to the right angle is called the hypotenuse.
The other two angles add to 90° and are called complementary angles.
The relationship between the sides and angles of a right triangle are given by the three basic trig relations which may be recalled with the mnemonic SOH-COH-TOA.
\begin{align*}
\sin\theta \amp= \frac{\textrm{opposite}}{\textrm{hypotenuse}} \amp \cos\theta \amp= \frac{\textrm{adjacent}}{\textrm{hypotenuse}} \amp \tan\theta \amp= \frac{\textrm{opposite}}{\textrm{adjacent}}
\end{align*}
and their inverses,
\begin{align*}
\theta \amp = \sin^{-1} \left(\frac{\textrm{opposite}}{\textrm{hypotenuse}}\right) \amp \theta \amp = \cos^{-1} \left(\frac{\textrm{adjacent}}{\textrm{hypotenuse}}\right) \amp \theta \amp = \tan^{-1} \left(\frac{\textrm{opposite}}{\textrm{adjacent}}\right)
\end{align*}
Facts.
The following statements regarding the trig functions and triangles are always true, and remembering them will help you avoid errors.
- \(\sin\text{,}\) \(\cos\) and \(\tan\) are functions of an angle and their values are unitless ratios of lengths.
- The inverse trig functions are functions of unitless ratios and their results are angles.
- The sine of an angle equals the cosine of its complement and vice-versa.
- The sine and cosine of any angle is always a unitless number between -1 and 1, inclusive.
- The sine, cosine, and tangent of angles between 0 and 90° are always positive.
- The inverse trig functions of positive numbers will always yield angles between 0 and 90°
- The legs of a right triangle are always shorter than the hypotenuse.
- Only right triangles have a hypotenuse.
Hints.
Here are some useful tips for angle calculations
- Take care that your calculator is set in degrees mode for this course.
- Always work with angles between 0° and 90° and use positive arguments for the inverse trig functions.
- Following this advice will avoid unwanted signs and incorrect directions caused because \(\dfrac{-a}{b} = \dfrac{a}{-b}\text{,}\) and \(\dfrac{a}{b} = \dfrac{-a}{-b}\) and the calculator can’t distinguish between them.
