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A comprehensive review of exercises related to trigonometric functions, including determining the amplitude, period, and phase shift of sine, cosine, and other trigonometric functions, and then graphing them over one complete period. The exercises cover a range of concepts, from basic sine and cosine functions to more complex trigonometric expressions. By working through these exercises, students can develop a deeper understanding of the properties and behaviors of trigonometric functions, which are fundamental in various areas of mathematics, physics, and engineering. The document serves as a valuable resource for students to practice and reinforce their knowledge of trigonometric functions, preparing them for more advanced topics and applications.
Typology: Exercises
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Exercise 1: Determine the amplitude and period of the given sine function and then graph it over
one complete period.
sin 3 3
y x
π = − (^)
Exercise 2: Determine the amplitude and period of the given cosine function and then graph it over
one complete period.
y =3cos 4 ( x )
Exercise 3: Determine the amplitude, period, and phase shift of the given trigonometric function
and then graph it over one complete period.
4 3sin 2 2
y x
Exercise 4: Graph the given trigonometric function over one complete period.
sin 3
y x
π = − (^) +
Exercise 5: Graph the given trigonometric function over one complete period.
y =2 cos 2 ( π x )
Find the y-values of the key points of the period
x-value y-value key point
0
( )
( )
( )
sin 0 3 3
sin 0 3
y
(0, 0)
( )
sin 3 3 4
sin 3 2
y
( )
( )
sin 3 3 2
sin 3
y
( )
sin 3 3 4
sin 3 2
y
x-value y-value key point
3
( )
( )
( )
sin 3 3 3
sin 2 3
y
(3, 0)
Graph the function
Exercise 2: Determine the amplitude and period of the given cosine function and then graph it over
one complete period.
y =3cos 4 ( x )
Find the amplitude
amplitude = A
x-value y-value key point
( )
3cos 4 8
3cos 2
y
( )
( )
3cos 4 4
3cos
y
π −
( )
3cos 4 8
3cos 2
y
( )
( )
3cos 4 2
3cos 2
y
Graph the function
Exercise 3: Determine the amplitude, period, and phase shift of the given trigonometric function
and then graph it over one complete period.
4 3sin 2 2
y x
Find the amplitude
4 3sin 2 2
sin 2 4 2
y x
y x
amplitude = A
Find the y-values of the key points of the period
x-value y-value key point
( )
( )
sin 2 4 4 2
sin 0 4
y
(^) π (^) π = (^) −
^
( )
sin 2 4 2 2
sin 4 2
y
( )
( )
sin 2 4 4 2
sin 4
y
π π
π
( )
( )
sin 2 4 2
sin 4 2
y
( )
( )
sin 2 4 4 2
sin 2 4
y
π π
π
Graph the function
Exercise 4: Graph the given trigonometric function over one complete period.
sin 3
y x
π = − (^) +
Find the amplitude
amplitude = A
Find the y-values of the key points of the period
x-value y-value key point
( )
sin 3
sin 3 3
sin 0
y x
sin 3
sin 6 3
sin 2
y x
( )
sin 3
sin 3 3
sin
y x
sin 3
sin 6 3
sin 2
y x
π
π π
π
x-value y-value key point
( )
sin 3
sin 3 3
sin 2
y x
π
Graph the function
Exercise 5: Graph the given trigonometric function over one complete period.
y =2 cos 2 ( π x )
Find the amplitude
amplitude = A
Find the period
period B
x-value y-value key point
1
2 cos 2 1
2 cos 2
(1, 2)
Graph the function
