“Right Triangle” Trigonometry

TK In addition to the coordinates of points on the unit circle, and in addition to percentages, the trigonometric functions can be thought of as returning the ratios of side-lengths of a right triangle with acute angles. \[ \sin\bigl(\theta\bigr) \!=\! \frac{\text{“opposite”}}{\text{“hypotenuse”}} \qquad \cos\bigl(\theta\bigr) \!=\! \frac{\text{“adjacent”}}{\text{“hypotenuse”}} \qquad \tan\bigl(\theta\bigr) \!=\! \frac{\text{“opposite”}}{\text{“adjacent”}} % \csc\bigl(\theta\bigr) \!=\! \frac{\text{“hypotenuse”}}{\text{“opposite”}} % \sec\bigl(\theta\bigr) \!=\! \frac{\text{“hypotenuse”}}{\text{“adjacent”}} % \cot\bigl(\theta\bigr) \!=\! \frac{\text{“adjacent”}}{\text{“opposite”}} \]

The other trigonometric functions secant and cosecant and cotangent can be defined in terms of sine and cosine.

\(\displaystyle \sec\bigl(\theta\bigr) \!=\! \frac{1}{\cos\bigl(\theta\bigr)}\)
\(\displaystyle \csc\bigl(\theta\bigr) \!=\! \frac{1}{\sin\bigl(\theta\bigr)}\)
\(\displaystyle \cot\bigl(\theta\bigr) \!=\! \frac{1}{\tan\bigl(\theta\bigr)}\)
\(\displaystyle \sin^2\bigl(\theta\bigr) \!+\! \cos^2\bigl(\theta\bigr) \!=\! 1\)
\(\displaystyle \tan^2\bigl(\theta\bigr) \!+\! 1 \!=\! \sec^2\bigl(\theta\bigr)\)
\(\displaystyle 1 \!+\! \cot^2\bigl(\theta\bigr) \!=\! \csc^2\bigl(\theta\bigr)\)

a 45-45-90 triangle a 30-60-90 triangle The three angles within any any triangle sum to 180°. The right triangles with angles 30°-60°-90° and 45°-45°-90° angles correspond to nice algebraic values of sine and cosine, and so their side-lengths are in nice ratios to each other. The supplement of an angle \(\theta\) is \(180°-\theta.\) The complement of an angle \(\theta\) is \(90°-\theta.\) The area of a triangle with an angle \(\theta\) with initial and terminal side lengths \(a\) and \(b\) is \(\frac{1}{2}ab\sin(\theta).\)