Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent
Use properties of even and odd trigonometric functions.
Recognize and use fundamental identities.
Evaluate trigonometric functions with a calculator.
Radio Tower Guy Wire Analysis
A radio transmission tower stands vertically with guy wires attached at various points to provide stability. We’ll use all six trigonometric functions to analyze the angles, distances, and tensions in these support structures.
A guy wire creates an angle whose terminal side passes through the point [latex]\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)[/latex] on the unit circle. Find all six trigonometric functions for this angle.
From the coordinates on the unit circle: [latex]\cos t = x = \frac{1}{2}[/latex] [latex]\sin t = y = \frac{\sqrt{3}}{2}[/latex] Now find the remaining four functions: [latex]\begin{aligned} \tan t &= \frac{\sin t}{\cos t} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3} \end{aligned}[/latex] [latex]\begin{aligned} \sec t &= \frac{1}{\cos t} = \frac{1}{\frac{1}{2}} = 2 \end{aligned}[/latex] [latex]\begin{aligned} \csc t &= \frac{1}{\sin t} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \end{aligned}[/latex] [latex]\begin{aligned} \cot t &= \frac{\cos t}{\sin t} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \cdot \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \end{aligned}[/latex]
[latex]\sin t = \frac{\sqrt{3}}{2}, \cos t = \frac{1}{2}, \tan t = \sqrt{3}, \sec t = 2, \csc t = \frac{2\sqrt{3}}{3}, \cot t = \frac{\sqrt{3}}{3}[/latex]
A measurement device records an angle of [latex]-\frac{\pi}{3}[/latex] radians. If [latex]\tan\left(\frac{\pi}{3}\right) = \sqrt{3}[/latex] and [latex]\sec\left(\frac{\pi}{3}\right) = 2[/latex], find [latex]\tan\left(-\frac{\pi}{3}\right)[/latex] and [latex]\sec\left(-\frac{\pi}{3}\right)[/latex].
Use the properties of even and odd functions: Even functions: [latex]\cos(-t) = \cos t[/latex] and [latex]\sec(-t) = \sec t[/latex] Odd functions: [latex]\sin(-t) = -\sin t[/latex], [latex]\tan(-t) = -\tan t[/latex], [latex]\csc(-t) = -\csc t[/latex], [latex]\cot(-t) = -\cot t[/latex] Since tangent is an odd function: [latex]\tan\left(-\frac{\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}[/latex] Since secant is an even function: [latex]\sec\left(-\frac{\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right) = 2[/latex]
A radio transmission tower stands vertically with guy wires attached at various points to provide stability. We’ll use all six trigonometric functions to analyze the angles, distances, and tensions in these support structures.