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78 | eUreka Math algebra II StUdy gUIde
● (^) Students learn the relationship among the constants A, ω, h, and k in the formula
fx()=-Axsin((w hk))+ and the properties of the sine graph. In particular, they
learn that
○ (^) |A| is the amplitude of the function. The amplitude is the distance between a maximal
point of the graph of the sinusoidal function and the midline (i.e., Af=-max k or A=ffmaxm- 2 in).
○ (^2) wp is the period of the function. The period P is the distance between two consecutive
maximal points (or two consecutive minimal points) on the graph of the sinusoidal
function. Thus, w=^2 Pp.
○ 2 wp is the frequency of the function (the frequency is the reciprocal of the period).
○ (^) h is called the phase shift.
○ (^) The graph of yk= is called the midline.
○ (^) Furthermore, the graph of the sinusoidal function f is obtained by vertically scaling the
graph of the sine function by A, horizontally scaling the resulting graph by w^1 , and then
horizontally and vertically translating the resulting graph by h and k units, respectively.
Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model Cyclical Behavior
● (^) Students review how changing the parameters A, ω, h, and k in
fx()=-Axsin((w hk))+
affects the graph of a sinusoidal function.
● (^) Students examine the example of the Ferris wheel, using height, distance from the
ground, period, and so on, to write a function of the height of the passenger cars in
terms of the sine function:
fx()=-Axsin((w hk))+
Lesson 13: Tides, Sound Waves, and Stock Markets
● (^) Students model cyclical phenomena from biological and physical science using
trigonometric functions.
● (^) Students understand that some periodic behavior is too complicated to be modeled by
simple trigonometric functions.
Lesson 14: Graphing the Tangent Function
● (^) Students graph the tangent function.
● (^) Students use the unit circle to express the values of the tangent function for p-x,
p+x, and 2 p-x in terms of tan(x), where x is any real number in the domain of the
tangent function.
Lesson 15: What Is a Trigonometric Identity?
● (^) Students prove the Pythagorean identity sinc^22 ()xx+=os () 1.
● (^) Students extend trigonometric identities to the real line, with attention to domain
and range.
● (^) Students use the Pythagorean identity to find sin (θ), cos (θ), or tan (θ), given sin (θ),
cos (θ), or tan (θ) and the quadrant of the terminal ray of the rotation.
Lesson 16: Proving Trigonometric Identities
● (^) Students prove simple identities involving the sine function, cosine function, and
secant function.
● (^) Students recognize features of proofs of identities.
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