Angles and circular motion are fundamental concepts in trigonometry. They help us understand how objects rotate and move in circles. We use degrees and radians to measure angles, and these measurements are crucial for calculating lengths and speeds.
The trigonometric circle ties it all together, showing how angles relate to points on a circle. This connection is key for grasping trigonometric functions and their applications in real-world scenarios, from engineering to physics.
Angles and Circular Motion
Angles in standard position
Vertex located at the origin with along positive x-axis
Formed by rotating counterclockwise from initial side
Positive angles created by counterclockwise rotation (90°)
Negative angles created by clockwise rotation (-45°)
Quadrantal angles have terminal side along coordinate axes
0° or 0 radians: points along positive x-axis
90° or 2π radians: points along positive y-axis
180° or π radians: points along negative x-axis
270° or 23π radians: points along negative y-axis
Degree and radian conversions
Degrees and radians both units for measuring angles
Convert degrees to radians by multiplying in degrees by 180π
60° = 60 × 180π = 3π radians
Convert radians to degrees by multiplying angle in radians by π180
4π radians = 4π × π180 = 45°
One complete revolution equals 360° or 2π radians
Coterminal angles and relationships
Angles sharing the same terminal side
Find in degrees by adding or subtracting multiples of 360°
45° and 405° coterminal (45° + 360° = 405°)
Find coterminal angles in radians by adding or subtracting multiples of 2π
6π and 613π coterminal (6π + 2π = 613π)
Arc length calculation
Arc is portion of circle's
Length depends on subtended angle at center (central angle) and circle's radius
formula: s=rθ (s = arc length, r = radius, θ = angle in radians)
Convert degrees to radians before using formula if needed
Circle with 4 unit radius and 90° subtended arc has length s=4×2π≈6.28 units
Circular motion speed concepts
Linear speed is rate object moves in straight line (ft/s)
is rate object rotates about fixed point (rad/s)
Relationship between linear speed (v), angular speed (ω), radius (r): v=rω
Point 2 meters from center with 3 rad/s angular speed has v=2×3=6 m/s linear speed
Period (t) is time for one complete rotation, frequency (f) is its reciprocal
f=t1
t=f1
Angular speed (ω) and frequency (f) related by ω=2πf
Angular Measurement and the Trigonometric Circle
Angular measurement is used to quantify the amount of rotation in circular motion
The trigonometric circle () is a fundamental tool for understanding angles and trigonometric functions
It has a radius of 1 unit and is centered at the origin of the coordinate plane
Points on the circle correspond to angle measures and their trigonometric function values
Key Terms to Review (23)
Angle: An angle is a measure of rotation between two intersecting lines or rays, typically measured in degrees or radians. It is fundamental in trigonometry for defining the sine and cosine functions.
Angular speed: Angular speed is the rate at which an object moves through an angle. It is usually measured in radians per second.
Arc: An arc is a portion of the circumference of a circle. It is defined by two points on the circle and the continuous part of the circle between them.
Arc length: Arc length is the measure of the distance along a section of the circumference of a circle. It is calculated using the formula $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.
Area of a sector: The area of a sector is the portion of the area of a circle bounded by two radii and their intercepted arc. It can be calculated using $A = \frac{1}{2} r^2 \theta$ when $\theta$ is in radians.
Circumference: The circumference is the distance around the edge of a circle. It can be calculated using the formula $C = 2\pi r$ or $C = \pi d$, where $r$ is the radius and $d$ is the diameter.
Coterminal angles: Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. They differ by an integer multiple of $360^\circ$ or $2\pi$ radians.
Degree: A degree is a unit of measurement for angles. It divides one full rotation into 360 equal parts.
Displacement: Displacement is the measure of the change in position of a point from one location to another. It is a vector quantity, meaning it has both magnitude and direction.
Endpoint: An endpoint is a point at the end of a line segment or the beginning and end points of an interval. They are crucial in determining the domain, range, and behavior of functions.
Initial side: The initial side of an angle is the fixed ray where the measurement of the angle begins. It is often positioned along the positive x-axis in standard position.
Measure of an angle: A measure of an angle indicates the size of the angle in degrees or radians. It quantifies the rotation needed to superimpose one of the rays on another.
Negative angle: A negative angle is an angle measured clockwise from the positive x-axis. It indicates the direction of rotation in a coordinate plane.
Positive angle: A positive angle is an angle measured counterclockwise from the initial side to the terminal side. It is typically measured in degrees or radians.
Quadrantal angle: A quadrantal angle is an angle in standard position whose terminal side lies on one of the coordinate axes. These angles are always multiples of $90^\circ$ or $\frac{\pi}{2}$ radians.
Radian: A radian is a unit of angular measure where the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. One full revolution around a circle is $2\pi$ radians.
Radian measure: Radian measure is a way of measuring angles based on the radius of a circle. An angle in radians is defined as the ratio of the length of the arc to the radius of the circle.
Ray: A ray is a part of a line that starts at a point and extends infinitely in one direction. It is used to represent angles and directions in trigonometric functions.
Reference angle: A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always between $0^{\circ}$ and $90^{\circ}$ inclusive.
Sector of a circle: A sector of a circle is a region bounded by two radii and the arc between them. It resembles a slice of pie or pizza.
Standard position: An angle is in standard position if its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. The angle's terminal side then determines its measure.
Terminal side: The terminal side of an angle is the ray where the angle ends, originating from the vertex and moving away from the initial side. It is used to determine the position of an angle in standard position on a coordinate plane.
Unit circle: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is used to define sine, cosine, and tangent functions for all real numbers.