In addition to the Sine, Cosine, and Tangent function, there are also what are known as the reciprocal trigonometric functions:
- Cosecant is the reciprocal of Sine: \(\csc(\theta) = \frac{{1}}{\sin(\theta)}\)
- Secant is the reciprocal of Cosine:\(\sec(\theta)=\frac{{1}}{\cos(\theta)}\)
- Cotangent is the reciprocal of Tangent: \(\cot(\theta) = \frac{{1}}{\tan(\theta)}\)
Example: Find the exact value of the trigonometric functions:
- \(\csc\left(\frac{7\pi}{{3}}\right)\)
- \(\sec\left(\frac{7\pi}{{3}}\right)\)
- \(\cot\left(\frac{7\pi}{{3}}\right)\)
Solution
\(\frac{7\pi}{{3}}\) is coterminal to \(\frac{\pi}{{3}}\), which can be calculated by subtracted \(2\pi\) from \(\frac{7\pi}{{3}}\): \(\frac{7\pi}{{3}}-2\pi=\frac{7\pi}{{3}}-\frac{6\pi}{{3}}-\frac{\pi}{{3}}\). Thus, each of the trigonometric functions above can be evaluated at \(\frac{\pi}{{3}}\) on the unit circle.
- \(\csc\left(\frac{7\pi}{{3}}\right)=\csc\left(\frac{\pi}{{3}}\right)=\frac{{1}}{\cos\left(\frac{\pi}{{3}}\right)}=\frac{{1}}{\frac{\sqrt{{3}}}{{2}}}=\frac{2\sqrt{ {{3}} }}{ {{3}} }\)
- \(\sec\left(\frac{7\pi}{{3}}\right)=\sec\left(\frac{\pi}{{3}}\right)=\frac{{1}}{\sin\left(\frac{\pi}{{3}}\right)}=\frac{{1}}{\frac{{1}}{{2}}}=2\)
- \(\cot\left(\frac{7\pi}{{3}}\right)=\cot\left(\frac{\pi}{{3}}\right)=\frac{{1}}{\tan\left(\frac{\pi}{{3}}\right)}=\frac{\frac{2\sqrt{{3}}}{{3}}}{{2}}=\frac{\sqrt{{3}}}{{3}}\)