The is a powerful tool for understanding trigonometric functions. It helps us visualize sine and cosine values for common angles like 30°, 45°, and 60°. These functions have specific domains and ranges, oscillating between -1 and 1.
Reference angles simplify trig calculations for angles outside the first quadrant. By relating any angle to its acute counterpart, we can easily determine function values. This concept is crucial for solving more complex trigonometric problems and understanding periodic behavior.
The Unit Circle and Trigonometric Functions
Sine and cosine for common angles
30° (6π radians)
Sine value is 21, which is the ratio of the opposite side to the hypotenuse in a 30-60-90 triangle
Cosine value is 23, which is the ratio of the adjacent side to the hypotenuse in a 30-60-90 triangle
45° (4π radians)
Both sine and cosine values are 22, which is the ratio of the leg to the hypotenuse in an isosceles right triangle
60° (3π radians)
Sine value is 23, which is the ratio of the opposite side to the hypotenuse in a 30-60-90 triangle
Cosine value is 21, which is the ratio of the adjacent side to the hypotenuse in a 30-60-90 triangle
Domain and range of trigonometric functions
Domain of both sine and cosine functions includes all real numbers (R), meaning the functions are defined for any input angle
Range of both sine and cosine functions is limited to the closed interval [−1,1]
Sine and cosine values oscillate between -1 and 1 as the angle varies
Maximum value of 1 occurs at 90° (2π radians) for sine and 0° (0 radians) for cosine
Minimum value of -1 occurs at 270° (23π radians) for sine and 180° (π radians) for cosine
Periodicity of sine and cosine functions is 360° or 2π radians, meaning the function values repeat every full rotation
Reference Angles and Their Applications
Reference angles for any angle
is the acute angle formed between the terminal side of the given angle and the x-axis
Calculation of reference angle depends on the quadrant of the given angle
Quadrant I: Reference angle is the same as the given angle
Quadrant II or III: Subtract the given angle from 180°
Reference angles simplify the evaluation of trigonometric functions for angles outside the first quadrant
Determine the absolute value of the function using the reference angle
Assign the appropriate sign based on the quadrant of the original angle
Quadrant I: All functions are positive
Quadrant II: Sine is positive, cosine is negative
Quadrant III: Tangent is positive, sine and cosine are negative
Quadrant IV: Cosine is positive, sine is negative
Example: Evaluating sin(120°)
Reference angle for 120° is 60°, and sin(60°) = 23
120° is in Quadrant II, where sine is positive
Therefore, sin(120°) = 23
Angle Measure and Standard Position
Angle measure refers to the quantity of rotation from the initial side to the terminal side of an angle
An angle is in standard position when its vertex is at the origin and its initial side is on the positive x-axis
Key Terms to Review (6)
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.
Cosine function: The cosine function, denoted as $\cos(\theta)$, is a trigonometric function that represents the x-coordinate of a point on the unit circle at an angle $\theta$ from the positive x-axis. It is periodic with a period of $2\pi$ and ranges from -1 to 1.
Pythagorean Identity: The Pythagorean Identity is a fundamental relation in trigonometry that states $\sin^2(\theta) + \cos^2(\theta) = 1$. It holds for any angle $\theta$ and is derived from the Pythagorean Theorem applied to the unit circle.
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.
Sine function: The sine function, denoted as $\sin(\theta)$, relates the angle $\theta$ in a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is a periodic function with a period of $2\pi$.
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.