Use reference angles, signs and definitions to determine exact values of trigonometric functions.
Subsection6.5.1Activities
Remark6.5.1.
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.
Definition6.5.2.
The unit circle is the circle of radius \(1\) centered at the origin on the coordinate plane.
Figure6.5.3.
Activity6.5.4.
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 the this context is the unit circle unless told otherwise.) We will label that point of intersection as \((x,y)\text{.}\)
(a)
What is the length of line segment \(r\text{,}\) whose endpoints are the origin and the point \((x,y)\) ?
\(\displaystyle 1\)
\(\displaystyle 2\)
\(\displaystyle 3\)
cannot be determined
Answer.
A
(b)
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.
Answer.
(c)
Using the triangle you’ve just created, find \(\cos \theta\text{.}\)
\(\displaystyle \dfrac{x}{y}\)
\(\displaystyle \dfrac{1}{x}\)
\(\displaystyle \dfrac{x}{1}\)
\(\displaystyle \dfrac{1}{y}\)
\(\displaystyle \dfrac{y}{1}\)
Answer.
C
(d)
Using that same triangle, find \(\sin \theta\text{.}\)
\(\displaystyle \dfrac{x}{y}\)
\(\displaystyle \dfrac{1}{x}\)
\(\displaystyle \dfrac{x}{1}\)
\(\displaystyle \dfrac{1}{y}\)
\(\displaystyle \dfrac{y}{1}\)
Answer.
E
(e)
Solve for \(x\) in one of the equations you’ve found above to determine an expression for the \(x\)-value of the point \((x,y)\) .
\(\displaystyle y\cos \theta\)
\(\displaystyle y\sin \theta\)
\(\displaystyle \cos \theta\)
\(\displaystyle \sin \theta\)
\(\displaystyle \dfrac{1}{\cos \theta}\)
\(\displaystyle \dfrac{1}{\sin \theta}\)
Answer.
C
(f)
Solve for \(y\) in one of the equations you’ve found above to determine an expression for the \(y\)-value of the point \((x,y)\) .
\(\displaystyle y\cos \theta\)
\(\displaystyle y\sin \theta\)
\(\displaystyle \cos \theta\)
\(\displaystyle \sin \theta\)
\(\displaystyle \dfrac{1}{\cos \theta}\)
\(\displaystyle \dfrac{1}{\sin \theta}\)
Answer.
D
Remark6.5.5.
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*}
Activity6.5.6.
Consider each angle \(\theta\) given below. Find the coordinates \((x,y)\) for the point at which \(\theta\) intersects the unit circle.
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.)
Activity6.5.8.
Let’s consider the angle \(\theta=150^\circ\text{,}\) drawn below with the unit circle.
(a)
What is the measure of the angle between the terminal side of \(\theta\) and the \(x\)-axis, as drawn below?
