Section2.3The Unit Circle and Trigonometric Functions
Definition2.3.1.
Alternate definition for trigonometric functions. In Figure 2.3.2, notice that since point \(A\) is on the unit circle, \(AB=1\text{.}\) Also, if \(A\) has coordinates \((x,y)\text{,}\) then \(BC=x\text{,}\) and \(AC=y\text{.}\) Use this information to rewrite \(\sin \theta\text{,}\)\(\cos \theta\text{,}\) and \(\tan \theta\) in terms of \(x\) and \(y\text{.}\) We can now define the three trigonometric functions for any angle \(\theta\) by measuring \(\theta\) as the angle (in radians) traversed counterclockwise from the positive \(x\)-axis.
Next, we will see how we can use special triangles and coordinates on the Unit Circle to compile a table of values of sine and cosine of commonly used angles.
First, use the unit circle Figure 2.3.4 to find the sine, cosine and tangent of 0, \(\frac{\pi}{2}\text{,}\)\(\pi\text{,}\)\(\frac{3 \pi}{2}\text{,}\)\(2\pi\) and record your answers in Table 2.3.5.
Perhaps because the ancient Greeks did a lot of geometry, it is common to use Greek letters in geometry, especially to label angles. In this case, the letters used are \(\alpha\) (alpha), \(\beta\) (beta), and \(\theta\) (theta). Also, recall that in a right triangle, we have that for any angle \(\phi\) (pronounced “fee") the definitions \(\sin \phi = \frac{opposite}{hypotenuse}\text{,}\)\(\cos \phi =\frac{adjacent}{hypotenuse}\text{,}\) and \(\tan \phi = \frac{opposite}{adjacent}\) (the acronym to remember these is SOHCAHTOA).
In Figure 2.3.8, \(\triangle ABD\) is an equilateral triangle with sides of length 1. Point \(C\) is the midpoint of \(\overline{AB}\text{.}\) Recall that the sum of the angles in a triangle is \(180^o\text{,}\) and that \(360^o\) is the same as \(2\pi\) radians. \(\triangle EFG\) is an isosceles right triangle, meaning that the two legs of the triangle are equal in length.
Use the two triangles above and the Unit Circle to determine the sine, cosine and tangent of the remaining angles in Table 2.3.5. Record your answers in the table.
While it is common to see the right triangle definitions for the trigonometric functions first, in practice, mathematicians often think of the equations in Definition 2.3.1 as the definition of the trig functions. The equations in Definition 2.3.1 lead to being able to graph the trigonometric functions for values beyond those in a right triangle, and later will be useful for modeling the motion of an object moving around in a circle.
At the url http://tinyurl.com/153sincos you will find an animation in Desmos that shows how the graphs of the sine and cosine function can be seen as functions with the angle with the \(x\)-axis as the input, and the coordinates of the point on the unit circle as the output.
Turn on the sine animation by clicking the circle next to ’Sine Animation’. Note how the graph of the sine function is traced out, with its values equal to the y-coordinate of points on the unit circle as the value of \(\alpha\) changes. Sketch the graph of \(f(x)=sin(x)\text{.}\)
Turn off the sine animation by clicking the circle next to ’Sine Animation’. Turn on the cosine animation by clicking the circle next to ’Cosine Animation’. Note how the graph of the cosine function is traced out, with its values equal to the x-coordinate of points on the unit circle as the value of \(\alpha\) changes. Sketch the graph of \(f(x)=cos(x)\text{.}\)
A Ferris wheel has a diameter of 24 feet, and is mounted so that the bottom of the wheel is 6 feet off the ground. The ride turns counterclockwise as seen by the operator.
