2ARecipTrigFuncs

Reciprocal Trig Functions

We know that for any right-angled triangle,

math \\ . \qquad \sin (\theta) = \dfrac {OPP}{HYP} \\. \\ . \qquad \cos (\theta) = \dfrac {ADJ}{HYP} \\. \\ . \qquad \tan \theta = \dfrac {OPP}{ADJ} math

The reciprocal of these ratios are defined as:

math \\ . \qquad \text{Cosecant:} \quad \;\; \text{cosec} (\theta) = \dfrac{1}{sin (\theta)} = \dfrac {HYP}{OPP} \\. \\ . \qquad \text{Secant:} \qquad \quad \;\; \text{sec} (\theta) = \dfrac{1}{cos (\theta)} = \dfrac {HYP}{ADJ} \\. \\ . \qquad \text{Cotangent:} \qquad \text{cot} (\theta) = \dfrac{1}{tan (\theta)} = \dfrac {ADJ}{OPP} math

**Note:** cosec is sometimes abbreviated to csc

The relationship can be remembered by noticing that the __third__ letter of the reciprocal function is always the initial letter of the original function.
 * co** s **ec is the reciprocal of ** s **in
 * se** c ** is the reciprocal of ** c **os
 * co** t ** is the reciprocal of ** t **an

Example 1

math \text{If } x = \dfrac{\pi}{3} math find the exact values for sec(x), cosec(x) and cot(x) First draw the triangle for the exact values of x.

math \text{cosec} \left( \dfrac{\pi}{3} \right) = \dfrac{1}{sin \left(\dfrac{\pi}{3} \right) } = \dfrac {HYP}{OPP}=\dfrac{2}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3} math

math \sec \left( \dfrac{\pi}{3} \right) = \dfrac{1}{cos \left( \dfrac{\pi}{3} \right) } = \dfrac {HYP}{ADJ}=\dfrac{2}{1}=2 math

math \cot \left( \dfrac{\pi}{3} \right) = \dfrac{1}{tan \left( \dfrac{\pi}{3} \right) } = \dfrac {ADJ}{OPP}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3} math

Example 2

Find the exact value for: math \cot \left( \dfrac {5\pi}{6} \right) math

math \tan \left( \dfrac {5\pi}{6} \right) = -\tan \left( \dfrac {\pi}{6} \right) = -\dfrac {1}{\sqrt{3}} math {tan is negative because in 2nd Quadrant}

math \cot \left( \dfrac {5\pi}{6} \right) = \dfrac {1}{\tan \left( \dfrac {5\pi}{6} \right) } = -\sqrt{3} math

Example 3

math \text {If cosec} \left( x \right) = \dfrac {4}{3} \text { and } 0 \leqslant x \leqslant 90^ \circ \text {, find } x \text{ (to the nearest tenth of a degree)} math

cosec(x) is the reciprocal of sin(x), so math \sin \left( x \right) = \dfrac {3}{4} math

Question is in degrees, so set calculator to degrees This is not an exact value, so using calculator, math \sin^{-1} \left( \dfrac {3}{4} \right) = 48.6^ \circ math

**See also:** www.mathsonline.com.au Y12Advanced --> Trigonometry --> Basic Trigonometry --> Lesson 4

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