The unit circle is a powerful tool for understanding trigonometric functions. It helps us visualize how sine and cosine values change as angles rotate around the circle. By memorizing key angles and their corresponding values, we can quickly solve trig problems.
Trigonometric functions have specific domains and ranges, which are crucial for graphing and solving equations. Reference angles simplify calculations for angles in different quadrants. Understanding these concepts allows us to tackle more complex problems involving angles and circular motion.
Unit Circle and Trigonometric Functions
Sine and cosine for common angles
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- Common angles in degrees and radians (30° = , 45° = , 60° = )
- Sine values for common angles (, , )
- Cosine values for common angles (, , )
- Memorize these values to quickly evaluate trigonometric functions for common angles without using a calculator
Domain and range of trigonometric functions
- Domain of sine and cosine functions includes all real numbers in radians or degrees
- Angle can be any value as it represents rotation around the unit circle
- Range of sine and cosine functions limited to values between -1 and 1, inclusive
- Unit circle has a radius of 1, sine and cosine values represent y-coordinate and x-coordinate of a point on the circle
- Understanding domain and range helps determine possible input and output values for trigonometric functions
Reference angles on unit circle
- Reference angle is the acute angle formed between terminal side of given angle and x-axis
- Finding reference angle depends on quadrant of given angle
- Quadrant I: reference angle same as given angle
- Quadrant II or III: subtract given angle from 180° or radians
- Quadrant IV: subtract 360° or radians from given angle
- Examples of finding reference angle (120° reference angle is 60°, reference angle is )
- Reference angles simplify evaluating trigonometric functions for angles in different quadrants
Trigonometric functions in all quadrants
- Evaluate sine and cosine for angles in different quadrants using reference angles and quadrant signs
- Determine reference angle
- Evaluate sine or cosine of reference angle
- Apply sign of function based on quadrant (Quadrant I: both positive, Quadrant II: sine positive, cosine negative, Quadrant III: both negative, Quadrant IV: sine negative, cosine positive)
- Example of evaluating
- Reference angle is 60°
- 120° in Quadrant II, sine is positive
- Therefore,
- Mastering trigonometric functions in all quadrants essential for solving complex problems involving angles and triangles
Unit Circle Properties and Motion
- The unit circle is centered at the origin (0,0) of the coordinate plane
- The radius of the unit circle is always 1 unit
- Circular motion around the unit circle represents the periodic nature of trigonometric functions
- The concept of periodicity in trigonometric functions is related to the complete rotation around the unit circle
- Angular velocity describes the rate of change of the angle as a point moves around the unit circle
Key Terms to Review (47)
Circle: A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.
Angle of rotation: The angle of rotation is the angle through which a figure or point is rotated about a fixed point, typically the origin. It is measured in degrees or radians.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Ellipse: An ellipse is a set of all points in a plane where the sum of the distances from two fixed points (foci) is constant. It is an important type of conic section.
Elimination: Elimination is a method for solving systems of equations where one variable is removed by adding or subtracting the equations. This process simplifies the system to a single-variable equation, which can then be solved.
Factor by grouping: Factor by grouping is a method used to factor polynomials that involves rearranging and combining terms into groups that have a common factor. This technique is particularly useful for polynomials with four or more terms.
Inequality: An inequality is a mathematical statement that indicates the relative size or order of two values using symbols such as <, >, ≤, or ≥. Inequalities can be solved to find the range of values that satisfy the given conditions.
Origin: In the rectangular coordinate system, the origin is the point where the x-axis and y-axis intersect. It is denoted by the coordinates (0, 0).
Nonlinear inequality: A nonlinear inequality is an inequality that involves a nonlinear function, such as quadratic, cubic, or higher-degree polynomials. These inequalities can be solved by finding the regions where the function is above or below a certain value.
System of nonlinear inequalities: A system of nonlinear inequalities consists of multiple inequalities involving nonlinear expressions. The solution is the set of points that satisfy all inequalities simultaneously.
X-coordinate: The x-coordinate is the first value in an ordered pair $(x, y)$ representing a point's horizontal position on the Cartesian plane. It indicates how far left or right the point is from the origin (0, 0).
System of nonlinear equations: A system of nonlinear equations consists of two or more equations with at least one equation that is not linear, i.e., it involves variables raised to a power other than one or products of variables.
(0, 1): (0, 1) is a point on the unit circle, which is a circle with a radius of 1 unit centered at the origin (0, 0) in the coordinate plane. The point (0, 1) represents the angle of 90 degrees or $\pi/2$ radians on the unit circle.
$[-1, 1]$: $[-1, 1]$ is a closed interval on the real number line that includes all real numbers between and including -1 and 1. This interval is commonly used in the context of the unit circle, which is a fundamental concept in trigonometry.
$ rac{ ext{sqrt}{3}}{2}$: $ rac{ ext{sqrt}{3}}{2}$ is a trigonometric ratio that represents the y-coordinate of a point on the unit circle when the angle is 30 degrees or $ rac{ ext{pi}}{6}$ radians. It is a fundamental value in trigonometry and is often used in the context of the unit circle to describe the behavior of sine and cosine functions.
$2\pi$: $2\pi$ is a fundamental mathematical constant that represents the circumference of a circle with a radius of 1 unit. It is an irrational number, meaning its decimal representation never repeats or terminates, and is approximately equal to 6.283185. This constant is crucial in understanding the behavior of periodic functions and the unit circle, which are central concepts in trigonometry and related mathematical fields.
$(– ext{infinity}, ext{infinity})$: $(– ext{infinity}, ext{infinity})$ refers to the set of all real numbers, including both positive and negative values, without any upper or lower bound. It represents the entire number line, extending from negative infinity to positive infinity.
$\frac{5\pi}{4}$: $\frac{5\pi}{4}$ is a specific angle measure on the unit circle, representing 5/4 of a full revolution or 450 degrees. This term is important in the context of the unit circle, as it helps describe the location and properties of points on the circle.
(-1, 0): The point (-1, 0) is a specific coordinate on the unit circle, which is a circle with a radius of 1 unit centered at the origin (0, 0) on the Cartesian coordinate plane. The coordinates (-1, 0) represent a point on the unit circle that corresponds to a specific angle and trigonometric function values.
$ rac{ ext{pi}}{6}$: $ rac{ ext{pi}}{6}$ is a fundamental angle measure in the unit circle, representing one-sixth of a full circle or 30 degrees. It is a significant angle in trigonometry and geometry, with various applications in mathematics and physics.
The term $ rac{ ext{pi}}{6}$ is closely tied to the unit circle, which is a circle with a radius of 1 unit, centered at the origin (0, 0) on the Cartesian coordinate plane. The unit circle is used to define and explore the trigonometric functions, such as sine, cosine, and tangent, and their relationships.
$ rac{pi}{3}$: $ rac{pi}{3}$ is a mathematical constant that represents one-third of the value of the mathematical constant pi (π), which is approximately equal to 3.14159. This term is particularly important in the context of the unit circle, as it represents a specific angle measure and the corresponding coordinates on the unit circle.
$ rac{ ext{sqrt}{2}}{2}$: $ rac{ ext{sqrt}{2}}{2}$ is a mathematical expression that represents a specific angle on the unit circle. It is the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle with one angle of 45 degrees.
(1, 0): (1, 0) is a point on the unit circle, which is a circle with radius 1 centered at the origin (0, 0) in the coordinate plane. This point represents one of the key reference points on the unit circle and is used to define and understand trigonometric functions.
$(- extinfinty°, extinfty°)$: $(- extinfinty°, extinfty°)$ refers to the set of all real numbers, including both positive and negative values, on the number line. This term is particularly relevant in the context of the unit circle, as it represents the range of possible angle measurements in degrees that can be represented on the unit circle.
$\frac{\pi}{4}$: $\frac{\pi}{4}$ is a fundamental mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number with a value of approximately 0.785398163, and it plays a crucial role in various mathematical and scientific applications, including the unit circle.
60°: 60° is an angle measurement that represents one-sixth of a full circle, or one-sixth of 360°. It is a fundamental angle in trigonometry and is commonly used in the context of the unit circle.
Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object. It describes the speed at which an object rotates or revolves around a fixed axis or point, and is a fundamental concept in the study of rotational motion.
30°: 30° is an angle that represents one-twelfth of a full circle, or 30 degrees. It is a fundamental angle used in the unit circle, a graphical representation of trigonometric functions.
45°: 45° is an angle that represents one-eighth of a full circle, or 1/8 of 360°. It is a fundamental angle in trigonometry and is often used as a reference point in the unit circle.
Circular Motion: Circular motion is the motion of an object in a circular path or orbit around a fixed point or axis. It is a fundamental concept in physics and mathematics, with applications in various fields such as astronomy, engineering, and everyday life.
Cosine: Cosine is one of the fundamental trigonometric functions, which describes the ratio between the adjacent side and the hypotenuse of a right triangle. It is a crucial concept in various areas of mathematics, including geometry, algebra, and calculus.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
Origin: The origin is a specific point in a coordinate system that serves as the reference point for all other points. It is the intersection of the x-axis and y-axis, and is typically denoted as the point (0, 0). The origin is a fundamental concept in various mathematical and scientific contexts, as it provides a common starting point for measurement and analysis.
Pi (π): Pi (π) is a fundamental mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends or repeats, and it is widely used in various mathematical and scientific applications, including the study of real numbers, angles, trigonometry, and the graphing of trigonometric functions.
Periodicity: Periodicity refers to the recurrence of a pattern or phenomenon at regular intervals. This concept is particularly relevant in the context of trigonometric functions, where it describes the repeated nature of the function's behavior over a specific domain.
Radian: A radian is a unit of angle measurement in mathematics, representing the angle subtended by an arc on a circle that is equal in length to the radius of that circle. It is a fundamental unit in trigonometry, providing a way to measure angles that is independent of the size of the circle.
Pythagorean Identity: The Pythagorean identity is a fundamental trigonometric identity that relates the trigonometric functions sine, cosine, and tangent. It is a crucial concept in understanding the unit circle and verifying, simplifying, and solving trigonometric expressions and equations.
Rotation: Rotation is the circular motion of an object around a fixed axis or point. It is a fundamental concept in mathematics and physics that describes the movement of an object as it turns around a central point or line.
Radius: The radius is the distance from the center of a circle to its perimeter. It is a fundamental measurement that defines the size and shape of a circular object.
Reference Angle: The reference angle is the acute angle formed between a given angle and the nearest coordinate axis in the unit circle. It provides a standardized way to represent and work with angles in the unit circle, allowing for easier calculations and comparisons.
Sine: The sine function, denoted as 'sin', is a trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions, along with cosine and tangent, and is essential in understanding various topics in college algebra.
SOHCAHTOA: SOHCAHTOA is a mnemonic device used to remember the trigonometric ratios of sine, cosine, and tangent in the context of right triangle trigonometry. It is also closely related to the unit circle and the geometric representation of trigonometric functions.
Terminal Side: The terminal side of an angle is the ray that forms the end or endpoint of the angle. It is one of the two rays that define the angle, with the other being the initial side.
Tangent: A tangent is a straight line that touches a curve at a single point, forming a right angle with the curve at that point. It is a fundamental concept in trigonometry and geometry, with applications across various mathematical disciplines.
Y-coordinate: The y-coordinate is the vertical position of a point on a coordinate plane, measured as the distance from the x-axis. It represents the up-down position of a point and is used to describe the location of objects or data points within a two-dimensional coordinate system.
X-Coordinate: The x-coordinate is the horizontal position of a point on a coordinate plane. It represents the distance from the origin (0,0) to the point along the horizontal x-axis. The x-coordinate is a crucial component in understanding and working with various mathematical concepts, including coordinate systems, graphs, unit circles, and systems of linear equations.
Quadrant: A quadrant is one of the four sections of the Cartesian coordinate plane, each defined by the signs of the coordinates of points located within it. These sections are important for understanding the positioning of angles and the values of trigonometric functions based on their reference angles. The concept of quadrants helps to visualize and categorize angles as they relate to circular functions and is crucial when applying reduction formulas.