📈College Algebra Unit 7 – The Unit Circle: Sine and Cosine

What's the Unit Circle?

• Defined as a circle with a radius of 1 centered at the origin (0, 0) on the x-y coordinate plane
• Circumference of the unit circle is $2\pi$ since the formula for circumference is $C = 2\pi r$ and the radius is 1
• Used to define trigonometric functions like sine, cosine, and tangent
• Helps to visualize and understand the relationship between angles and their corresponding trigonometric values
• Angles are measured in radians, where one full rotation is equal to $2\pi$ radians or 360 degrees
• Positive angle rotations move counterclockwise around the circle, while negative rotations move clockwise
• Any point on the unit circle can be represented as $(x, y) = (\cos\theta, \sin\theta)$, where $\theta$ is the angle formed with the positive x-axis

Key Points on the Unit Circle

• The x-coordinate of any point on the unit circle represents the cosine of the angle formed with the positive x-axis
• The y-coordinate of any point on the unit circle represents the sine of the angle formed with the positive x-axis
• The unit circle is divided into four quadrants, labeled I, II, III, and IV, starting from the top-right and moving counterclockwise
• Quadrant I: both sine and cosine are positive
• Quadrant II: sine is positive, and cosine is negative
• Quadrant III: both sine and cosine are negative
• Quadrant IV: sine is negative, and cosine is positive
• At 0 radians (0°), $(x, y) = (1, 0)$, meaning $\cos(0) = 1$ and $\sin(0) = 0$
• At $\frac{\pi}{2}$ radians (90°), $(x, y) = (0, 1)$, meaning $\cos(\frac{\pi}{2}) = 0$ and $\sin(\frac{\pi}{2}) = 1$

Sine and Cosine Basics

• Sine and cosine are trigonometric functions that relate angles to the x and y coordinates on the unit circle
• For an angle $\theta$, $\sin(\theta)$ is equal to the y-coordinate, and $\cos(\theta)$ is equal to the x-coordinate of the point where the angle intersects the unit circle
• Sine and cosine are periodic functions, meaning they repeat their values at regular intervals
  • The period of sine and cosine is $2\pi$ radians or 360 degrees
• The range of both sine and cosine is [-1, 1], meaning the output values are always between -1 and 1, inclusive
• Sine and cosine are odd and even functions, respectively
  • For an odd function like sine, $\sin(-\theta) = -\sin(\theta)$
  • For an even function like cosine, $\cos(-\theta) = \cos(\theta)$
• The relationship between sine and cosine can be expressed using the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

Connecting Angles to Coordinates

• To find the coordinates of a point on the unit circle given an angle, use the formulas $x = \cos(\theta)$ and $y = \sin(\theta)$
  • For example, if $\theta = \frac{\pi}{3}$, then $x = \cos(\frac{\pi}{3}) = \frac{1}{2}$ and $y = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
• To find the angle given the coordinates of a point on the unit circle, use the inverse trigonometric functions arcsine (sin⁻¹) and arccosine (cos⁻¹)
  • For example, if $(x, y) = (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$, then $\theta = \arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4}$ or $\theta = \arcsin(-\frac{\sqrt{2}}{2}) = -\frac{\pi}{4}$
• When using inverse trigonometric functions, pay attention to the quadrant of the point to determine the correct angle
• Remember that arcsine returns angles in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$, while arccosine returns angles in the range $[0, \pi]$

Special Angles and Their Values

• There are several commonly used angles in the unit circle that have exact trigonometric values
• 0 radians (0°): $\sin(0) = 0$, $\cos(0) = 1$
• $\frac{\pi}{6}$ radians (30°): $\sin(\frac{\pi}{6}) = \frac{1}{2}$, $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$
• $\frac{\pi}{4}$ radians (45°): $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
• $\frac{\pi}{3}$ radians (60°): $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$, $\cos(\frac{\pi}{3}) = \frac{1}{2}$
• $\frac{\pi}{2}$ radians (90°): $\sin(\frac{\pi}{2}) = 1$, $\cos(\frac{\pi}{2}) = 0$
• Memorizing these special angles and their sine and cosine values can help you quickly solve problems involving the unit circle
• You can also use these values to find the trigonometric values for angles in other quadrants by using symmetry and the properties of odd and even functions

Graphing Sine and Cosine

• The graphs of sine and cosine are smooth, continuous curves that oscillate between -1 and 1
• The sine graph starts at the origin, while the cosine graph starts at (0, 1)
• Both graphs have a period of $2\pi$, meaning they repeat every $2\pi$ units
• The amplitude of both graphs is 1, which is the distance from the midline to the maximum or minimum point
• The midline of both graphs is the x-axis (y = 0)
• To graph sine and cosine functions with transformations, use the general forms:
  • $y = a \sin(b(x - c)) + d$
  • $y = a \cos(b(x - c)) + d$
  • $a$ represents the amplitude
  • $b$ represents the frequency (how often the graph repeats)
  • $c$ represents the phase shift (horizontal shift)
  • $d$ represents the vertical shift
• Understanding the effects of these transformations on the graphs of sine and cosine is crucial for modeling periodic phenomena

Real-World Applications

• Trigonometric functions, particularly sine and cosine, have numerous real-world applications
• In physics, sine and cosine are used to model simple harmonic motion, such as the motion of a pendulum or a spring
• In engineering, sine and cosine are used to analyze and design electrical circuits, particularly in the study of alternating current (AC)
• In acoustics, sine and cosine functions are used to represent sound waves and analyze their properties, such as frequency and amplitude
• In navigation, sine and cosine are used to calculate distances and angles on the Earth's surface, which is essential for GPS systems
• In computer graphics, sine and cosine functions are used to create smooth, realistic animations and to rotate and transform objects in 2D and 3D space
• Understanding the properties and applications of sine and cosine functions is crucial for success in various fields, including science, technology, engineering, and mathematics (STEM)

Common Mistakes and How to Avoid Them

• Confusing radians and degrees: Always pay attention to the angle units and convert between radians and degrees when necessary using the formula $\theta_{radians} = \frac{\pi}{180} \times \theta_{degrees}$
• Mixing up sine and cosine: Remember that sine corresponds to the y-coordinate, and cosine corresponds to the x-coordinate on the unit circle
• Forgetting the negative signs in quadrants II, III, and IV: Keep in mind the signs of sine and cosine in each quadrant (I: ++, II: -+, III: --, IV: +-)
• Incorrectly applying the Pythagorean identity: The Pythagorean identity is $\sin^2(\theta) + \cos^2(\theta) = 1$, not $\sin(\theta) + \cos(\theta) = 1$
• Misinterpreting the period of sine and cosine: The period is $2\pi$ radians or 360 degrees, not $\pi$ radians or 180 degrees
• Incorrectly using inverse trigonometric functions: Pay attention to the quadrant when using arcsine or arccosine to find the angle, and remember the range of angles each function returns
• Not considering the domain and range of sine and cosine: The domain is all real numbers, while the range is [-1, 1]
• By being aware of these common mistakes and taking steps to avoid them, you can improve your understanding and application of sine, cosine, and the unit circle in College Algebra and beyond