Recall the radian measure is defined by the equation:
- θ = s/r
- *s & r must be the same units so ratio is a real number without units
an equation satisfied by all values of the variable in its domain
Name the Pythagorean Identities (3) *not trig identities
- sin2t + cos2t = 1
- tan2t + 1 = sec2t
- cot2t + 1 = csc2t
A function f is said to be an even function provided:
also state an example
- f(-t) = f(t) for every value of t in the domain of f
- f(t) = t2
The graph of an even function is always symmetric about the ______. Whether we use t or -t in f(t) = t2, the ______is the same
A function f is said to be an odd function provided:
State an example
- f(-t) = -f(t) for every value of t in the domain of f
- g(t) = t3
In g(t) = t3, if you compute g(5) and g(-5), you'll find that the outputs are the _______ of one another. The graph of an odd function is always symmetric about the ______.
The domains of both even and odd functions rely on which property?
Must have the property that if t is in the domain, then -t is also in the domain.
The Opposite-Angle Identities
- cos(-t) = cos t
- sin(-t) = -sin t
- tan(-t) = -tan t
Imagine a line connecting P to Q along with an arc (t) extending from x-axis to P and another arc (-t) extending from x-axis to Q. What are the x and y coordinates for both P and Q and define tan(-t) in terms of sin(-t) & cos(-t)
- P: (cos t, sin t)
- Q: (cos (-t), sin (-t))
- tan (-t) = sin(-t)/cos(-t) = -sin t/cos t
- -(sin t/cos t) = -tan t
The final identities in 7.1 are simply consequences of the fact that circumference C of the unit circle is ___ (explain). Thus if we begin at any point P on the unit circle and travel a distance of ____ units along the perimeter, we return to the same point P (explain).
- 2π (substitute r = 1 in the formula C = 2πr gives us C = 2π)
- arc lengths of t and t + 2π (1,0) yield the same terminal point on the unit circle (1,0).
Periodicity of all trig functions (6)
- sin(t + 2π) = sin t
- cos(t + 2π) = cos t
- tan(t + 2π) = tan t
- csc(t + 2π) = csc t
- sec(t + 2π) = sec t
- cot(t + 2π) = cot t
Explain the periodicity functions
Basically, if we start at a point P on the unit circle and make two complete counterclockwise revolutions, the arc length we travel is 2π + 2π = 4π.
For 3 complete revolutions the arc length traversed is 3(2π) = 6π. In general, if k is any integer, the arc length for k complete revolutions is 2|k|π. When K is positive, the revolutions are _________; when k is negative, the revolutions are ________. What are the two resulting identities (sin & cos)
- sin(t + 2kπ) = sin t
- cos(t + 2kπ) = cos t
Functions whose graphs display patterns that repeat themselves at regular intervals
The period of a function represents the ______ _______ of units that we must travel along the ______ _____ before the graph begins to _____ itself.
- minimum number
- horizontal axis
A nonconstant function f is said to be periodic if:
there is a number p>0 such that f(x + p) = f(x) for all x in the domain of f. The smallest such number p is called the period of f
For a function in which the graph is centered about the horizontal axis, the amplitude is simply the _______ ______ of the graph above the _______ ______.
- maximum height
- horizontal axis
The amplitude of any periodic function (w/ formula)
- Let f be a periodic function and let me and M denote, respectively, the minimum and maximum values of the function. Then the amplitude of f is the number
List the θ and sin θ values possible, as though you were about to make a very detailed graph for y = sin θ
- θ: 0, π,/6, π/3, π/2, 2π/3, 5π/6, π, 7π/6, 4π/3, 3π/2, 5π/3, 11π/6 & 2π
- sin θ: 0, 1/2, √3/2, 1, √3/2, 1/2, 0, -1/2, -√3/2, -1, -√3/2, -1/2 & 0
Note: the graph of y = sin x on an interval of length ___, the period of the sine function, is called a _____ of the graph of y = sin x
The domain of the sine function is the set of ____ ____ ______. The range of the (general) sine function is the ______ interval of _____.
- all real numbers
- closed interval
- -1≤ sin x ≤1 for all x
The sine function an _____ periodic function with period of _____ and an amplitude of ____
The graph of y = sin x consists of repetitions, over the entire domain of the basic sine wave (figure 10 pg 513). Where does the basic sine wave cross the x-axis? Where does it peak? Where is its lowest point?
- Beginning, middle and end
- one quarter of the way through the cycle
- three quarters of the way through the cycle
Which identity of cos x is most helpful when attempting to graph y = cos x
cos x = sin (x + π/2)
The graph of y = cos x is obtained by translating the sine curve _____ units over to the ____.
The domain of the cosine function is the set of ___ ____ _____. The range of the cosine function is the ______ interval _______.
- all real numbers
- closed interval
- -1≤ cos x ≤1 for all x
The cosine function is an ____ periodic function with period ____. The amplitude is ____.
The graph of y = cos x consists of repetitions, over the entire domain, of the basic cosine wave. Where does the basic cosine wave cross the x axis, and where are its highest and lowest points?
- One quarter & three quarters of the way through the basic cycle.
- The curve peaks at the beginning and the end of the basic cycle and reaches it lowest point half way through the basic cycle
Use the reference angle concept to find all solutions to the equations cos x = .8 in the interval 0≤ x ≤2π
- Treat as though someone asked for the cos of 45 degrees and that -315 degrees
- x2 = 2π - x1= 2π - cos-1approx = 5.640
cos-1 (x) denotes the unique number in the interval [0,π] whose cosine is x
state 3 examples demonstrating this (explain each)
- cos-1 (1/2) = π/3 *because cos π/3 = 1/2 and 0<π/3<π
- cos-1 (1/2) ≠ 5π/3 *because although cos 5π/3 = 1/2, the number 5ππ/3 is not in the required interval [0,π]
- cos-1 (-1/2) = 2π/3 *because cos 2π/3 = -1/2 and 0<2π/3<π
- cos-1 (0.8) ≃ 0.644 *(example 2 pg 517)
sin-1: sin-1(x) is the unique number in the interval [-π/2, π/2]
cos x = .8 find x
cos inverse of both sides
Use the reference angle concept and a calculator to find all solutions of the equation cos x = .8 in the interval 0≤x≤2π. Which formula do you use?
- x2 = 2π - x1
- x2 = 2π - cos-1(.8)
Use the reference angle concept and a calculator to find all solutions of the equations cos x = -.8 within the interval 0≤x≤2π. Which formulas do you use? (explain)
- x2 = π - x1 = π - cos-1(.8)
- x2 = π + x1 = π +cos-1(.8)
- *because cosine is negative in quadrants II and III, we want angles in Quadrants II and III that have have x1 for the reference angle. *Check figure 16 pg 517 for the proof that the equations are legit
How do you obtain the graph of y = 2 sinx from that of y = sinx. How does this affect the amplitude and the period?
- Multipy each y-coordinate on the graph of y = sin x by 2.
- Amplitude changes from 1 to 2
- Period remains at 2π
How do you change y = cos x into y = 1/2 cos x. How does this affect the amplitude and the period.
- Multiply each y-coordinate on the graph of y = cos x by 1/2
- Amplitude becomes 1/2 instead of 1
- Period remains 2π
Generally, graphs of functions of the form y = A sin x and y = A cos x always have an amplitude of _____ and a period of ____.
How do you turn y = sin x into y = -sin x?
A reflection about the x axis (crude flip)
How would you graph the function y = cos 3x over one period.
- Note* cosine curve y = cos x begins its basic pattern when x = 0 and ends when x = 2π
- ∴ y = cos 3x will begin at 3x = 0 and end at 3x = 2π (x = 2π/3)
- 2π/3 becomes our new period, which we will divide into 4 quarters: 0, π,/6, π/3, π/2, 2π/3 (figures obtained from the more detailed list in card 20)
- Now graph with amplitude of 1
How would you define the amplitude and period of (for example) y = A sin Bx in light of the previous card?
- Amplitude = |A|
- period = 2π/B
How would you figure out the period of a cos or sin function knowing that 3/4th of way to the end of the cycle, we have the coordinates (9,-4)
- 2π/B = period
- (3/4)2π/B = 9
- (4/3)(3/4)(2π/B) = (9)(4/3)
- 2π/B = 12
- 2π = 12B
- B = π/6
What is the first thing to do when asked to graph the function y = 4 sin (2x -2π/3) over one period
- First factor out the 2 within the parentheses: y = 4 sin[2(x - π/3)] *we want to do this with the simplest translations
- Next, graph y = 4 sin 2x and like any other function, translate the to right by π/3 units.
What is the exact formula for the phase shift π/3 in y = 4sin(2x - 2π/3)
- y = A sin (Bx - C)
- C/B = (-2π/3)/2 = -π/3 aka to the right π/3 units