Trigonometry / Unit 3: The Unit Circle and Trig Functions of Any Angle
3.5 Reciprocal Functions and Quadrantal Angles
Notes
Find a reciprocal function by flipping the value of its partner.
- $\displaystyle \csc\theta = \frac{1}{\sin\theta}$, $\displaystyle \sec\theta = \frac{1}{\cos\theta}$, $\displaystyle \cot\theta = \frac{1}{\tan\theta}$.
- Example: $\displaystyle \sec 30^\circ = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$.
- If the partner is 0, the reciprocal is undefined.
- At quadrantal angles: $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ$ is undefined, $\cos 180^\circ = -1$, $\sin 270^\circ = -1$.
Practice
What is $\sec 60^\circ$?
$\displaystyle \cos 60^\circ = \frac{1}{2}$, so $\sec 60^\circ = 2$.
What is $\csc 210^\circ$?
$\displaystyle \sin 210^\circ = -\frac{1}{2}$, so $\csc 210^\circ = -2$.
What is $\tan 90^\circ$?
$\displaystyle \tan 90^\circ = \frac{\sin 90^\circ}{\cos 90^\circ} = \frac{1}{0}$, which is undefined.
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