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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
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An identity
an equation satisfied by all values of the variable in its domain
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Name the Pythagorean Identities (3) *not trig identities
- sin2t + cos2t = 1
- tan2t + 1 = sec2t
- cot2t + 1 = csc2t
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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
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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
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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
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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 ______.
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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.
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The Opposite-Angle Identities
- cos(-t) = cos t
- sin(-t) = -sin t
- tan(-t) = -tan t
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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
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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)
- 2π
- arc lengths of t and t + 2π (1,0) yield the same terminal point on the unit circle (1,0).
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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
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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.
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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)
- counterclockwise
- clockwise
- sin(t + 2kπ) = sin t
- cos(t + 2kπ) = cos t
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Periodic function
Functions whose graphs display patterns that repeat themselves at regular intervals
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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
- repeat
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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
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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
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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
- (M-m)/2
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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
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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
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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
- [-1,1]
- -1≤ sin x ≤1 for all x
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The sine function an _____ periodic function with period of _____ and an amplitude of ____
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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
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Which identity of cos x is most helpful when attempting to graph y = cos x
cos x = sin (x + π/2)
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The graph of y = cos x is obtained by translating the sine curve _____ units over to the ____.
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The domain of the cosine function is the set of ___ ____ _____. The range of the cosine function is the ______ interval _______.
- all real numbers
- closed interval
- [-1,1]
- -1≤ cos x ≤1 for all x
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The cosine function is an ____ periodic function with period ____. The amplitude is ____.
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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
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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
- so:
- x2 = 2π - x1= 2π - cos-1approx = 5.640
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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)
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Inverse sine
sin-1: sin-1(x) is the unique number in the interval [-π/2, π/2]
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cos x = .8 find x
cos inverse of both sides
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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)
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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
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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
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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
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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 ____.
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How do you turn y = sin x into y = -sin x?
A reflection about the x axis (crude flip)
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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
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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
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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
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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.
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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
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