Trigonometry
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Table of Contents
- The Unit Circle and Angle Measure
- The Unit Circle
- Angle Measure in Radians
- Angle Measure in Degrees
- Arc Length and Sector Area
- The Trigonometric Functions
- Sine and Cosine Functions
- Key Angles in the Unit Circle
- Reference Angles and the Rest of the Unit Circle
The Unit Circle
[justify]We define a circle as the set of points on a plane that are equidistant from a point [math]\left(x_{0,} \ y_0\right)[/math], which is the center of the circle. The distance from each point of the circle to the center is called the radius. The equation of a circle centered at [math]\left(x_{0,} \ y_0\right)[/math] with radius [math]r[/math] is [/justify][center][math] \large \left(x-x_0\right)^2 - \left( y - y_0\right)^2 = r^2. [/math] [/center][br]The [b]unit circle[/b] is the circle centered at the origin with radius of 1 unit, so the equation of the [b]unit circle[/b] is[br][br][center][math] \huge x^2+y^2=1.[/math][/center][br]The unit circle is shown below. The point on the unit circle can be moved, showing the approximate coordinates. Note the right triangle formed by mapping out the coordinates. Note that the equation [math]x^2+y^2=1[/math] is simply an application of the [url=https://tube.geogebra.org/material/simple/id/222063]Pythagorean theorem[/url].
Interactive Unit Circle
Sine and Cosine Functions
[justify]On the unit circle, we define the sine of an angle [math]\theta[/math], denoted [math] \sin \theta[/math], as the [math]y[/math]-coordinate of the terminal point of [math] \theta[/math] and the cosine of [math] \theta[/math], denoted [math] \cos \theta[/math] as the [math]x[/math]-coordinate of the terminal point of [math] \theta[/math]. Recall that the unit circle is the graph of the equation [math] x^2+y^2=1[/math]. By substituting [math] x = \cos \theta[/math] and [math] y = \sin \theta,[/math] we immediately have the Pythagorean identity[/justify][center][math]\Large \cos^2 \theta + \sin^2 \theta = 1.[/math][/center]Since the arc length is the same as the angle subtended on the unit circle, [math] \theta[/math] can be considered an angle or a real number corresponding to the arc length on the unit circle starting from the point [math](1,\,0)[/math]. Counter-clockwise rotation is considered a positive direction of rotation, and clockwise is negative.
Sine and Cosine in the Unit Circle
[justify]In circles of arbitrary radius [math]r,[/math] the arc length, [math]x,[/math] and [math]y[/math] coordinates all grow or shrink proportionally to [i][math]r.[/math] [/i]So the coordinates of the terminal point [i][math]P[/math] [/i]are [math] \large x = r \cos \theta[/math] and [math] \large y = r \sin \theta[/math]. Dividing both sides of both equations by [math]r[/math] gives the more general definitions for sine and cosine for a circle of any radius.[/justify][center][math]\Large \cos \theta = \frac{x}{r}, \ \ \sin \theta = \frac{y}{r}[/math][/center][br]
Sine and Cosine in Circle of Radius r
[justify]The circle of radius [i][math]r[/math][/i] centered at the origin is defined by the equation [math] x^2 + y^2 = r^2[/math]. Substituting for [i][math]x[/math][/i] and [i][math]y[/math][/i] we have [math] (r \cos \theta)^2 + (r \sin \theta )^2 = r^2[/math]. Dividing both sides by [math] r^2[/math], we see that the Pythagorean identity [math] \cos^2 \theta + \sin^2 \theta = 1[/math] holds for a circle of any radius. [/justify]