All Flashcards
What is the unit circle?
A circle with a radius of 1, centered at the origin (0,0).
What is an angle in standard position?
An angle that starts from the positive x-axis and goes counterclockwise.
Define 'radians'.
A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius.
Define 'terminal ray'.
A line from the origin at a given angle that intersects the unit circle.
Define sine (sin θ) in the context of the unit circle.
The y-coordinate of the point where the terminal ray intersects the unit circle.
Define cosine (cos θ) in the context of the unit circle.
The x-coordinate of the point where the terminal ray intersects the unit circle.
Define tangent (tan θ) in the context of the unit circle.
The ratio of the y-coordinate to the x-coordinate (sin θ / cos θ) of the point where the terminal ray intersects the unit circle.
What are reflex angles?
Angles greater than 180° but less than 360°.
What is a 'full rotation' in radians?
2π radians.
What is the ASTC rule?
A mnemonic to remember which trig functions are positive in each quadrant: All, Sine, Tangent, Cosine.
What are the differences between sine and cosine functions?
Sine: y-coordinate on the unit circle, odd function | Cosine: x-coordinate on the unit circle, even function
What are the differences between positive and negative angles on the unit circle?
Positive angles: Measured counterclockwise | Negative angles: Measured clockwise
What are the differences between radians and degrees?
Radians: Based on the radius of the circle, dimensionless | Degrees: Arbitrary division of a circle into 360 parts
Compare sin(θ) and cos(θ) at θ = 0.
sin(0) = 0 | cos(0) = 1
Compare sin(θ) and cos(θ) at θ = π/2.
sin(π/2) = 1 | cos(π/2) = 0
Compare the range of sine and cosine functions.
Sine: [-1, 1] | Cosine: [-1, 1]
Compare the graphs of y = sin(x) and y = cos(x).
y = sin(x): Starts at (0, 0), odd function | y = cos(x): Starts at (0, 1), even function
Compare the derivatives of sin(x) and cos(x).
Derivative of sin(x): cos(x) | Derivative of cos(x): -sin(x)
Compare the integrals of sin(x) and cos(x).
Integral of sin(x): -cos(x) + C | Integral of cos(x): sin(x) + C
Compare the use of sine and cosine in right triangles.
Sine: Opposite/Hypotenuse | Cosine: Adjacent/Hypotenuse
Explain the relationship between the unit circle and trigonometric functions.
The unit circle provides a visual representation of trigonometric functions, where the x and y coordinates of points on the circle correspond to cosine and sine values of angles, respectively.
Explain how to find the sine and cosine of an angle using the unit circle.
Locate the angle on the unit circle. The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.
Explain the significance of the radius being 1 in the unit circle.
It simplifies trigonometric calculations and directly relates angles to trigonometric values without needing to account for scaling.
How do positive and negative angles differ on the unit circle?
Positive angles are measured counterclockwise from the positive x-axis, while negative angles are measured clockwise.
Explain how to determine the sign of trigonometric functions in different quadrants.
Use the ASTC rule: All functions are positive in Quadrant I, Sine is positive in Quadrant II, Tangent is positive in Quadrant III, and Cosine is positive in Quadrant IV.
Explain the concept of coterminal angles.
Coterminal angles are angles that share the same terminal side when drawn in standard position. They differ by multiples of 2π radians or 360 degrees.
Why are trigonometric ratios constant for a given angle?
Regardless of the size of the triangle, the ratios of sides for a given angle remain constant, as defined by the unit circle.
How does the unit circle help in understanding periodic functions?
The cyclical nature of the unit circle illustrates the periodic behavior of sine and cosine functions, repeating every 2π radians.
Explain how to use the unit circle to find angles given sine or cosine values.
Locate the y-coordinate for sine or x-coordinate for cosine on the unit circle and identify the corresponding angle(s).
What is the relationship between sine and cosine values at complementary angles?
Sine of an angle is equal to the cosine of its complement, and vice versa (e.g., sin(θ) = cos(90° - θ)).