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 guy wire is attached to the tower at an angle of [latex]\frac{5\pi}{4}[/latex] radians from the positive x-axis. Find all six trigonometric functions for this angle.
First, determine the quadrant. Since [latex]\frac{5\pi}{4}[/latex] is between [latex]\pi[/latex] and [latex]\frac{3\pi}{2}[/latex], the angle is in Quadrant III. Find the reference angle: [latex]\begin{aligned} \text{Reference angle} &= \frac{5\pi}{4} - \pi \ &= \frac{5\pi}{4} - \frac{4\pi}{4} \ &= \frac{\pi}{4} \end{aligned}[/latex] For the reference angle [latex]\frac{\pi}{4}[/latex], we know: [latex]\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}[/latex] In Quadrant III, both x and y coordinates are negative, so both sine and cosine are negative. However, when both are negative, tangent and cotangent become positive (negative divided by negative).
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]
If [latex]\sin t = -\frac{3}{5}[/latex] and [latex]t[/latex] is in Quadrant III, find the values of the other five trigonometric functions.
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.