📐Algebra and Trigonometry Unit 7 – The Unit Circle: Sine and Cosine

Key Concepts

• The unit circle is a circle with a radius of 1 centered at the origin (0, 0) on the Cartesian plane
• Angles in the unit circle are measured in radians, where one full rotation equals $2\pi$ radians
• Sine and cosine are trigonometric functions that relate angles to the x and y coordinates on the unit circle
• Special angles ($0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$) have specific sine and cosine values that can be derived using the unit circle
  • These values are often used as reference points for solving trigonometric problems
• Sine and cosine functions are periodic, meaning they repeat their values at regular intervals
  • The period of both functions is $2\pi$
• Graphing sine and cosine functions requires understanding their amplitude, period, phase shift, and vertical shift
• Real-world applications of the unit circle and trigonometric functions include modeling periodic phenomena (sound waves, tides) and calculating distances and angles in navigation and surveying

The Unit Circle Defined

• The unit circle is a circle with a radius of 1 unit centered at the origin (0, 0) on the Cartesian plane
• The circle's circumference is divided into 2π radians, with angles measured counterclockwise from the positive x-axis
• The x and y coordinates of any point on the unit circle can be determined using the cosine and sine of the angle formed by the positive x-axis and the line connecting the origin to that point
  • The x-coordinate is equal to the cosine of the angle
  • The y-coordinate is equal to the sine of the angle
• The unit circle is a fundamental tool for understanding and visualizing trigonometric functions
• Memorizing the unit circle and the corresponding sine and cosine values for special angles is essential for solving trigonometric problems efficiently
• The unit circle can be divided into four quadrants, each with its own set of angle measures and corresponding sine and cosine values
• Angles in the unit circle can be positive or negative, depending on the direction of rotation (counterclockwise or clockwise)

Angles and Radians

• Angles in the unit circle are measured in radians, where one full rotation equals $2\pi$ radians
• To convert degrees to radians, multiply the angle in degrees by $\frac{\pi}{180}$
• To convert radians to degrees, multiply the angle in radians by $\frac{180}{\pi}$
• Radians provide a more natural way to measure angles in the unit circle, as they are based on the circle's radius rather than an arbitrary division of a circle into 360 degrees
• Understanding the relationship between radians and the unit circle is crucial for working with trigonometric functions
  • For example, the sine of an angle in radians is equal to the y-coordinate of the corresponding point on the unit circle
• Angles in the unit circle can be represented as multiples of π (e.g., $\frac{\pi}{4}, \frac{3\pi}{2}$) or as decimal values (e.g., 0.785, 4.712)
• Negative angle measures in radians represent clockwise rotations, while positive angle measures represent counterclockwise rotations

Sine and Cosine on the Unit Circle

• Sine and cosine are trigonometric functions that relate angles to the x and y coordinates on the unit circle
• The sine of an angle is equal to the y-coordinate of the corresponding point on the unit circle
  • Sine is abbreviated as sin, so for an angle θ, the sine is written as $\sin(\theta)$
• The cosine of an angle is equal to the x-coordinate of the corresponding point on the unit circle
  • Cosine is abbreviated as cos, so for an angle θ, the cosine is written as $\cos(\theta)$
• The sine and cosine values for any angle can be determined using the unit circle and the x and y coordinates of the corresponding point
• Sine and cosine are periodic functions, meaning they repeat their values at regular intervals
  • The period of both functions is $2\pi$, which corresponds to one full rotation around the unit circle
• The range of the sine function is [-1, 1], while the range of the cosine function is also [-1, 1]
  • These ranges represent the minimum and maximum y-coordinates and x-coordinates, respectively, on the unit circle
• Understanding the relationship between sine, cosine, and the unit circle is essential for solving trigonometric equations and graphing trigonometric functions

Special Angles and Their Values

• Special angles ($0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$) have specific sine and cosine values that can be derived using the unit circle
• These values are often used as reference points for solving trigonometric problems and graphing trigonometric functions
• The sine and cosine values for these special angles are:
  • $\sin(0) = 0$, $\cos(0) = 1$
  • $\sin(\frac{\pi}{6}) = \frac{1}{2}$, $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$
  • $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
  • $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$, $\cos(\frac{\pi}{3}) = \frac{1}{2}$
  • $\sin(\frac{\pi}{2}) = 1$, $\cos(\frac{\pi}{2}) = 0$
• Memorizing these values can help simplify trigonometric calculations and make it easier to identify patterns and relationships between angles and their sine and cosine values
• Special angles can be used to determine the sine and cosine values for other angles through the use of trigonometric identities and symmetry properties of the unit circle
• The sine and cosine values for special angles in the second, third, and fourth quadrants can be determined using the values from the first quadrant and the sign conventions for each quadrant

Graphing Sine and Cosine Functions

• Graphing sine and cosine functions requires understanding their amplitude, period, phase shift, and vertical shift
• The general form of a sine function is $y = a \sin(b(x - c)) + d$, where:
  • $a$ represents the amplitude (half the distance between the maximum and minimum values)
  • $b$ represents the frequency (related to the period by $b = \frac{2\pi}{period}$)
  • $c$ represents the phase shift (horizontal shift of the graph)
  • $d$ represents the vertical shift (vertical shift of the graph)
• The general form of a cosine function is $y = a \cos(b(x - c)) + d$, with the same parameters as the sine function
• The default sine and cosine functions ($y = \sin(x)$ and $y = \cos(x)$) have an amplitude of 1, a period of $2\pi$, no phase shift, and no vertical shift
• To graph a sine or cosine function, start by plotting the key points using the special angles and their corresponding sine or cosine values
  • Then, connect these points with a smooth curve, taking into account the amplitude, period, phase shift, and vertical shift
• Sine and cosine functions are often used to model periodic phenomena in the real world, such as sound waves, tides, and seasonal temperature variations
• Understanding how to graph and interpret sine and cosine functions is crucial for solving problems in various fields, including physics, engineering, and economics

Real-World Applications

• The unit circle and trigonometric functions have numerous real-world applications in various fields
• In physics, sine and cosine functions are used to model periodic motion (pendulums, springs) and wave phenomena (sound waves, light waves)
  • For example, the displacement of a pendulum over time can be modeled using a sine function
• In engineering, trigonometric functions are used to analyze and design mechanical systems (gears, motors) and electrical systems (alternating current)
  • For instance, the voltage in an AC circuit can be represented using a cosine function
• In navigation and surveying, the unit circle and trigonometric functions are used to calculate distances, angles, and positions
  • For example, the distance between two points on the Earth's surface can be determined using the great-circle distance formula, which involves the cosine function
• In computer graphics and animation, trigonometric functions are used to create realistic motion and 3D transformations (rotations, scaling)
  • For instance, the rotation of an object in 3D space can be achieved using sine and cosine functions
• In economics and finance, trigonometric functions are used to model periodic trends and cycles in data (stock prices, economic indicators)
  • For example, the seasonal variation in sales data can be modeled using a combination of sine and cosine functions
• Understanding the real-world applications of the unit circle and trigonometric functions can help students appreciate the relevance and importance of these concepts in various fields

Common Mistakes and How to Avoid Them

• Confusing radians and degrees when working with angles in the unit circle
  • Always pay attention to the angle units and convert between radians and degrees when necessary
• Forgetting the sign conventions for sine and cosine values in different quadrants of the unit circle
  • Remember: All Students Take Calculus (ASTC) - Quadrant I: All positive, Quadrant II: Sine positive, Quadrant III: Tangent positive, Quadrant IV: Cosine positive
• Mixing up the sine and cosine values for special angles
  • Create a mnemonic device or visual aid to help remember the sine and cosine values for special angles
• Incorrectly applying the amplitude, period, phase shift, or vertical shift when graphing sine and cosine functions
  • Double-check the general form of the function and the values of the parameters before graphing
• Neglecting to consider the domain and range of sine and cosine functions when solving problems
  • Remember that the domain of both functions is all real numbers, while the range is [-1, 1]
• Making arithmetic errors when simplifying trigonometric expressions or solving equations
  • Take your time and show your work step-by-step to minimize the risk of errors
• Failing to recognize when trigonometric identities or properties can be applied to simplify a problem
  • Familiarize yourself with common trigonometric identities and properties, and practice identifying when they can be used
• Not checking the reasonableness of your answers in the context of the problem
  • Always take a moment to assess whether your answer makes sense given the information provided in the problem