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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)
Explain the concept of periodicity in trigonometric functions.
Trigonometric functions repeat their values at regular intervals. Sine and cosine repeat every 2π radians, while tangent repeats every π radians.
Explain the relationship between sine, cosine, and the unit circle.
On the unit circle, sin(θ) is the y-coordinate and cos(θ) is the x-coordinate of a point on the circle corresponding to the angle θ.
Explain why radians are preferred over degrees in calculus.
Radians are based on the geometry of the circle (arc length to radius ratio), making them more natural in mathematical and physical contexts compared to the arbitrary division of degrees.
What happens to tan(θ) when cos(θ) = 0?
tan(θ) is undefined when cos(θ) = 0, which occurs at odd multiples of π/2.
Describe the behavior of sine in the first quadrant.
Sine increases from 0 to 1 as the angle increases from 0 to π/2.
Describe the behavior of cosine in the first quadrant.
Cosine decreases from 1 to 0 as the angle increases from 0 to π/2.
Describe the behavior of tangent in the first quadrant.
Tangent increases from 0 to infinity as the angle increases from 0 to π/2.
How does the sign of sine change in different quadrants?
Sine is positive in quadrants I and II, and negative in quadrants III and IV.
How does the sign of cosine change in different quadrants?
Cosine is positive in quadrants I and IV, and negative in quadrants II and III.
How does the sign of tangent change in different quadrants?
Tangent is positive in quadrants I and III, and negative in quadrants II and IV.