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Question 1 of 4
Which graph best represents both of the trigonometric functions below?
y = cos x y = cos x
y = sec x y = sec x
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First, identify the values of the cos cos function
y y
= =
a a cos cos b b x x
y y
= =
a cos x a cos x
Next, solve for the period of the cos cos function
P cos P cos
= =
2 π b 2 π b
= =
2 π 1 2 π 1
Substitute known values
= =
2 π 2 π
To graph the cos cos curve, it is better to divide the value of its period into for parts and have the curve meet the following conditions
Curve starts at the peak of the amplitude ( a ) ( a )
Curve intercepts x-axis at 1st quarter
Curve reaches minimum amplitude at 2nd quarter
Curve intercepts x-axis again at 3rd quarter
Curve starts again at the period (P = 2 π P = 2 π )
Therefore, this will be the cos cos curve for y = sin x y = sin x with a period of 2 π 2 π
Next, recall that sec x = 1 cos x sec x = 1 cos x
Given this, we can graph y = csc x y = csc x with regards to the following:
= =
Undefined values of 1 cos x will be asymptotes Undefined values of 1 cos x will be asymptotes
= =
Defined values of 1 cos x will be points of intersection Defined values of 1 cos x will be points of intersection
Finally, graph y = sec x y = sec x
Asymptotes
1 cos π 2 1 cos π 2
= =
undefined undefined
1 cos 3 π 2 1 cos 3 π 2
= =
undefined undefined
Points of intersection
1 cos 0 1 cos 0
= =
1 1
1 cos π 1 cos π
= =
- 1 − 1
1 cos 2 π 1 cos 2 π
= =
1 1
Take note that the curves of y = sec x y = sec x will keep approaching but will never intersect with the asymptotes.
Question 2 of 4
Graph the trigonometric function
y = 3 cos 2 x y = 3 cos 2 x
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First, identify the values of the function
y y
= =
a a cos cos b b x x
y y
= =
3 cos 2 x 3 cos 2 x
Next, solve for the period of the function
P cos P cos
= =
2 π b 2 π b
= =
2 π 2 2 π 2
Substitute known values
= =
π π
To graph the cos cos curve, it is better to divide the value of its period into for parts and have the curve meet the following conditions
Curve starts at the peak of the amplitude ( a ) ( a )
Curve intercepts x-axis at 1st quarter
Curve reaches minimum amplitude at 2nd quarter
Curve intercepts x-axis again at 3rd quarter
Curve starts again at the period (P = π P = π )
Therefore, this will be the cos cos curve for y = 3 cos 2 x y = 3 cos 2 x with a period of π π
Question 3 of 4
Graph the trigonometric function
y = 4 cos 2 3 x y = 4 cos 2 3 x
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First, identify the values of the function
y y
= =
a a cos cos b b x x
y y
= =
4 cos 2 3 x 4 cos 2 3 x
Next, solve for the period of the function
P cos P cos
= =
2 π b 2 π b
= =
2 π 2 3 2 π 2 3
Substitute known values
= =
6 π 2 6 π 2
= =
3 π 3 π
To graph the cos cos curve, it is better to divide the value of its period into for parts and have the curve meet the following conditions
Curve starts at the peak of the amplitude ( a ) ( a )
Curve intercepts x-axis at 1st quarter
Curve reaches minimum amplitude at 2nd quarter
Curve intercepts x-axis again at 3rd quarter
Curve starts again at the period (P = 3 π P = 3 π )
Therefore, this will be the cos cos curve for y = 4 cos 2 3 x y = 4 cos 2 3 x with a period of 3 π 3 π
Question 4 of 4
Graph the trigonometric function within the domain - π 2 ≤ x ≤ 2 π − π 2 ≤ x ≤ 2 π
y = tan x y = tan x
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First, solve for the period of the function
P tan P tan
= =
π b π b
= =
π 1 π 1
Substitute known values
= =
π π
Asymptotes will occur within the domain for the values of tan tan that are undefined
tan - π 2 tan − π 2
= =
undefined undefined
tan π 2 tan π 2
= =
undefined undefined
tan 3 π 2 tan 3 π 2
= =
undefined undefined
To graph the tan tan curve, start at the lower domain ( - π 2 ) ( − π 2 ) and draw a curve going passing through ( 0 , 0 ) ( 0 , 0 ) and heading up to the second asymptote ( π 2 ) ( π 2 ) . This will cover a period of the curve.
Repeat the process until you reach the upper domain ( 2 π ) .