In Section 6.4, we learned how to find the exact values of the six trigonometric ratios for the special acute angles \(30^\circ\text{,}\)\(45^\circ\text{,}\) and \(60^\circ\text{.}\) In this section, we will use that knowledge and expand to finding the exact trig values of any multiple of those angles.
Let \(\theta\) be the angle shown below in standard form. Notice that the terminal side intersects with the unit circle. (Note: We will assume a circle drawn in this context is the unit circle unless told otherwise.) We will label that point of intersection as \((x,y)\text{.}\)
We will now create a right triangle using the previous line segment \(r\) as the hypotenuse. Draw in a line segment of length \(x\) and another of length \(y\) to create such a triangle.
From the previous activity, we have found a connection between the sine and cosine values of an angle \(\theta\) and the coordinates \((x,y)\) of the point at which that angle intersects the unit circle. Namely,
\begin{equation*}
x=\cos \theta \, \text{ and }\, y = \sin \theta
\end{equation*}
In Activity 6.5.6, we found \((x,y)\)-coordinates (and thus the sine and cosine values) for angles that terminated either in Quadrant 1 or on an axis adjacent to Quadrant 1. We’ll now expand to angles that terminate elsewhere, using our knowledge of the cosine and sine values of angles in the first quadrant along with how reflections over the \(x\) and \(y\) axes affect the signs of the coordinates. (See Section 2.4 for a reminder on how these reflections work.)
We can find the sine and cosines values of our original angle, \(\theta=150^\circ\text{,}\) by using the angle \(\alpha=30^\circ\) to help. Find the point \((x_1,y_1)\text{,}\) where the terminal side of the \(30^\circ\) angle intersects the unit circle.
How does the point you’ve just found compare with the point \((x,y)\text{,}\) where the terminal edge of \(\theta=150^\circ\) intersects the unit circle?
The reference angle for a given angle \(\theta\) is the angle in the first quadrant obtained from reflecting \(\theta\text{.}\) Equivalently, it is the smallest angle between the terminal side of \(\theta\) and the \(x\)-axis.
The angle below represents the reference angle for \(\theta=\dfrac{4\pi}{3}\text{,}\) which is the smallest angle between the terminal side of \(\theta\) and the \(x\)-axis. What is the measure of this reference angle?
We can find the sine and cosines values of our original angle, \(\theta=\dfrac{4\pi}{3}\text{,}\) by using the reference angle to help. Find the point \((x_1,y_1)\text{,}\) where the terminal side of the angle \(\dfrac{\pi}{3}\) intersects the unit circle.
How does the point you’ve just found compare with the point \((x,y)\text{,}\) where the terminal edge of \(\theta=\dfrac{4\pi}{3}\) intersects the unit circle?
So far we’ve only dealt with angles that are part of a special right triangle (30-60-90 or 45-45-90) or are a multiple of one of these angles, but we can extend to other angles as well.
Let \(\theta\) be the angle whose terminal side intersects the unit circle at the point described in each situation below. Find \(\sin \theta\text{,}\)\(\cos \theta\text{,}\)\(\tan \theta\text{,}\)\(\sec \theta\text{,}\)\(\csc \theta\text{,}\) and \(\cot \theta\text{.}\)
