All Flashcards
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.
Define 'standard position' of an angle.
Vertex at the origin, initial ray along the positive x-axis.
What are coterminal angles?
Angles sharing the same terminal ray, differing by multiples of 360° (or 2π radians).
Define a radian.
Angle measure based on the arc length subtended on a circle; ratio of arc length to radius.
Define sine (sin θ).
Ratio of the y-coordinate of a point on the unit circle to the radius: sin(θ) = y/r.
Define cosine (cos θ).
Ratio of the x-coordinate of a point on the unit circle to the radius: cos(θ) = x/r.
Define tangent (tan θ).
Slope of the terminal ray; ratio of the y-coordinate to the x-coordinate: tan(θ) = y/x.
Alternative definition of tangent (tan θ).
tan(θ) = sin(θ) / cos(θ)
What is the range of sine?
[-1, 1]
What is the range of cosine?
[-1, 1]
What is the range of tangent?
All real numbers.
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.