13. The equation of the given function is [tex]y=4sin[\frac{(t-\frac{4}{3})}{2}]-2[/tex]
14. The equation of the cotangent function is [tex]y=cot[2(t-\frac{1}{3})]+2[/tex]
Step-by-step explanation:
Let us revise the transformation of the trigonometric function:
y = a f[b(x + c)] + d, where
- Amplitude is a
- f represents the trigonometry function
- Period is 2π/b
- Phase shift is c (positive is to the left)
- Vertical shift is d
13.
∵ y = a sin( [tex]\frac{2\pi t}{T}[/tex] ), where a is the amplitude
and T is the wave in seconds
∵ The amplitude is 4
∴ a = 4
∵ The period is 4π
∴ T = 4π
From the rules above
∵ The period is 2π/b
∴ T = [tex]\frac{2\pi }{B}[/tex]
∴ 4π = [tex]\frac{2\pi }{b}[/tex]
- By using cross multiplication
∴ 4π(b) = 2π
- Divide both sides by 4π
∴ b = [tex]\frac{1}{2}[/tex]
∵ The phase shift is [tex]-\frac{4}{3}\pi[/tex]
∵ c is the phase shift
∴ c = [tex]-\frac{4}{3}\pi[/tex]
∵ The vertical shift is -2
∵ d is the vertical shift
∴ d = -2
Now substitutes the values of a, b, c and d in the form of the equation below
∵ y = a sin[b(t + c)] + d
∴ [tex]y=4sin[\frac{1}{2}(t-\frac{4}{3})]-2[/tex]
You can write it as [tex]y=4sin[\frac{(t-\frac{4}{3})}{2}]-2[/tex]
The equation of the given function is [tex]y=4sin[\frac{(t-\frac{4}{3})}{2}]-2[/tex]
14.
y = cot[b(t + c)] + d
∵ The period = π
∵ The period is 2π/b
- Equate π by 2π/b to find b
∴ π = [tex]\frac{2\pi }{b}[/tex]
- By using cross multiplication
∴ π(b) = 2π
- Divide both sides by π
∴ b = 2
∵ The phase shift is [tex]-\frac{1}{3}\pi[/tex]
∵ c is the phase shift
∴ c = [tex]-\frac{1}{3}\pi[/tex]
∵ The vertical shift is 2
∵ d is the vertical shift
∴ d = 2
Now substitutes the values of b, c and d in the form of the equation below
∵ y = cot[b(t + c)] + d
∴ [tex]y=cot[2(t-\frac{1}{3})]+2[/tex]
The equation of the cotangent function is [tex]y=cot[2(t-\frac{1}{3})]+2[/tex]
Learn more:
You can learn more about trigonometry function in brainly.com/question/3568205
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