- Know the trigonometric function values for the special angles in radians
- Know the trigonometric function values for the special angles in radians
- Use a unit circle to find trig values
- Find reference angles in radians
- Evaluate trigonometric expressions
- Find coordinates on a unit circle
- Find an angle with a given terminal point on a unit circle
- Use the tangent ratio to find slope
- Find coordinates on a circle of radius
Measuring angles in radians has other applications besides calculating arclength, and we will need to evaluate trigonometric functions of angles in radians. The sine, cosine, or tangent of a particular angle is the same whether the angle is measured in radians or in degrees.
For example, [latex]\dfrac{\pi}{3}[/latex] radians is the same as [latex]60°{,}[/latex] because [latex]\dfrac{\pi}{3} \cdot \dfrac{180°}{\pi} = 60°{,}[/latex] so
[latex]\sin \dfrac{\pi}{3} = \sin 60° = \dfrac{\sqrt{3}}{2}[/latex]
However, we don’t have to convert radians to degrees in order to evaluate trig ratios; your calculator can give you the trigonometric function values for angles expressed in radians. First, change the calculator setting from Degree mode to Radian mode. Then enter
[latex]\qquad\qquad\qquad[/latex]SIN [latex]\dfrac{\pi}{3}[/latex]
and the calculator will return [latex]0.8660254038{.}[/latex] You can check that this number is a decimal approximation for [latex]\dfrac{\sqrt{3}}{2}{.}[/latex]
Use your calculator to find the sine and cosine of the following angles in radians. Round your answers to four decimal places.
- [latex]\theta = \dfrac{7\pi}{4}[/latex]
- [latex]\theta = 3.5[/latex]
Solution
- With your calculator in radian mode, enter COS [latex]7\pi[/latex] ÷ [latex]4[/latex] and SIN [latex]7\pi[/latex] ÷ [latex]4[/latex] to find
[latex]\cos (7\pi/4) = 0.7071067812\\ \sin (7\pi/4) = -0.7071067812[/latex]
Rounding to four decimal places gives [latex]\cos \dfrac{7\pi}{4} = 0.7071[/latex] and [latex]\sin \dfrac{7\pi}{4} = -0.7071{.}[/latex]
- Your calculator will give you the values
[latex]\cos (3.5) = -0.9364566873\\ \sin (3.5) = -0.3507832277[/latex]
Rounding to four places, we have [latex]\cos (3.5) = -0.9365[/latex] and [latex]\sin (3.5) = -0.3508{.}[/latex] Note that 3.5 radians is a third-quadrant angle, so the signs of the trig values make sense.
Checkpoint 6.17.
Use your calculator to find the tangents of the following angles in radians. Round your answers to four decimal places.
- [latex]\theta = \dfrac{5\pi}{12}[/latex]
- [latex]\theta = 5.2[/latex]
Solution
- [latex]3.7321[/latex]
- [latex]-1.8856[/latex]
Coterminal angles measured in radians have the same trig values, just as they do when measured in degrees. So adding or subtracting a multiple of [latex]2 \pi[/latex] to any angle results in a new angle in the same standard position.
Find the sine and cosine of the following angles in radians. Round your answers to four decimal places.
- [latex]\theta = \dfrac{5\pi}{2}[/latex]
- [latex]\theta = - 4[/latex]
Solution
- Because [latex]\dfrac{5\pi}{2} = 2\pi + \dfrac{\pi}{2},~[/latex] [l