All Flashcards
What does the graph of y = sin(x) tell us about its derivative?
The derivative, cos(x), represents the slope of the sine function. Where sine increases, cosine is positive; where sine decreases, cosine is negative; where sine has a max/min, cosine is zero.
What does the graph of y = cos(x) tell us about its derivative?
The derivative, -sin(x), represents the slope of the cosine function. Where cosine increases, -sine is positive; where cosine decreases, -sine is negative; where cosine has a max/min, -sine is zero.
What does the graph of y = tan(x) tell us about its derivative?
The derivative, sec²(x), is always positive, indicating that the tangent function is always increasing (except where it's undefined).
How does the amplitude affect the graph of sine and cosine?
The amplitude determines the maximum and minimum values of the function. A larger amplitude stretches the graph vertically.
How does the period affect the graph of sine, cosine, and tangent?
The period determines how often the function repeats. A shorter period compresses the graph horizontally, while a longer period stretches it.
What does a vertical shift in the graph of sine or cosine indicate?
A vertical shift indicates a change in the midline of the function. The entire graph is shifted up or down.
What does a horizontal shift in the graph of sine or cosine indicate?
A horizontal shift (phase shift) indicates a change in the starting point of the function. The entire graph is shifted left or right.
How can you identify the period of a trigonometric function from its graph?
The period is the distance along the x-axis required for the function to complete one full cycle.
How can you identify the amplitude of a sine or cosine function from its graph?
The amplitude is half the distance between the maximum and minimum values of the function.
What does the graph of sin(2x) look like compared to sin(x)?
The graph of sin(2x) is a horizontal compression of sin(x) by a factor of 2, meaning it oscillates twice as fast.
How do you convert an angle from degrees to radians?
Multiply the angle in degrees by π/180.
How do you find a coterminal angle?
Add or subtract multiples of 360° (or 2π radians) to the given angle.
Given sin(θ) and the quadrant, how do you find cos(θ)?
Use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to find the absolute value of cos(θ), then determine the sign based on the quadrant.
Given cos(θ) and the quadrant, how do you find sin(θ)?
Use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to find the absolute value of sin(θ), then determine the sign based on the quadrant.
How to find tan(θ) if you know sin(θ) and cos(θ)?
Divide sin(θ) by cos(θ): tan(θ) = sin(θ) / cos(θ).
How do you determine the period of y = sin(bx) or y = cos(bx)?
Period = 2π / |b|.
How do you determine the period of y = tan(bx)?
Period = π / |b|.
How do you solve for x in sin(x) = a, where -1 <= a <= 1?
Find the principal value using arcsin(a), then use the properties of sine to find other solutions in the desired interval.
How do you solve for x in cos(x) = a, where -1 <= a <= 1?
Find the principal value using arccos(a), then use the properties of cosine to find other solutions in the desired interval.
How do you solve for x in tan(x) = a?
Find the principal value using arctan(a), then add multiples of π to find other solutions in the desired interval.
What are the differences between sine and cosine?
Sine starts at 0 at x=0, cosine starts at 1 at x=0 | Sine is an odd function, cosine is an even function
Compare the graphs of sine and cosine.
Sine is a wave that passes through the origin, cosine is a wave that starts at its maximum value | Sine and cosine have the same period and amplitude but are shifted horizontally by π/2.
What are the key differences between tangent and sine/cosine?
Tangent has a period of π, sine/cosine have a period of 2π | Tangent has vertical asymptotes, sine/cosine are continuous.
Compare the ranges of sine/cosine and tangent.
Sine and cosine have a range of [-1, 1] | Tangent has a range of all real numbers.
Contrast the behavior of sine and cosine in the first quadrant.
Sine increases from 0 to 1 | Cosine decreases from 1 to 0
Compare the points where sine and cosine are zero.
Sine is zero at multiples of π | Cosine is zero at odd multiples of π/2
Compare the derivatives of sine and cosine.
The derivative of sine is cosine | The derivative of cosine is -sine
What is the difference between positive and negative angles?
Positive angles are measured counterclockwise | Negative angles are measured clockwise
Compare radian and degree measure.
Radians are based on the radius of a circle | Degrees are an arbitrary division of a circle into 360 parts
Compare the graphs of y = sin(x) and y = sin(-x).
y = sin(-x) is a reflection of y = sin(x) across the x-axis | sin(x) is an odd function, so sin(-x) = -sin(x)