Q: A function generator produces $$V = 2\sin(2\alpha t) - 2.5\cos(\alpha t) - 2.5. If$$ $$\alpha = 43$$, which trigonometric substitution is used to convert $$\sin(2\alpha t)$$ into a function of $$\cos(\alpha t)$$?

$$\sin(2\theta) = 2\sin \theta \cos \theta$$

Q: If the resultant voltage $$V_4$$ has rectangular components $$x_{total} = 410.388$$ and $$y_{total} = 178.238$$, what is the phase angle $$\theta (in $$radians) of $$V_4$$?

$$\theta = \tan$$^{-1}$$\left(\frac{178.238}{410.388}\right)$$

Question 1

Given the voltage function $$V_1 = A$$ \sin(0.5t)$$, and knowing that at t = 5$ seconds, V_1$ = 10.75 V$$, what is the value of $$A$$? (Assume all angles are in radians.)

Accepted Answer

$$A = 17.96$$

Question 2

Which trigonometric identity is used to expand $$V_3 = 36$$\sin(0.5t + C)$$ into a sum of sine and cosine terms?

Accepted Answer

$$\sin(A+B) = \sin A \cos B + \cos A \sin B$$

Question 3

Given $$V_1 = A$$\sin(0.5t)$$ and V_2$ = B\cos(0.5t)$$, and $$V_3$$ = $$V_1$$ + $$V_2 = 36$$\sin(0.5t + C)$$, what is the value of B$ if C = 1.048$ radians?

Accepted Answer

$$B = 36\sin(1.048) \approx 31.19$$

Question 4

If $$V = 2\sin(2\alpha t) - 2.5\cos(\alpha t) - 2.5$$ and $$\alpha = 43$$, what is the first value of $$t (in $$seconds) when $$V = 0$$?

Accepted Answer

$$t = 0.0424$$

Question 5

A BJT switch requires at least $$2.5 V to $$turn on. If the voltage signal $$V$$ completes a cycle every $$0.2 s $$and the rollercoaster is in the camera's frame for $$1 s$$, how many rows of seats can be photographed?

Accepted Answer

$$5$$

Question 6

Given $$V_1 = 443$$\sin(\omega t)$$, V_2$ = 250\sin(\omega t + 2\pi/3)$$, and $$V_3 = 100$$\sin(\omega t - \pi/8)$$, what is the rectangular (Cartesian) form of V_2$?

Accepted Answer

$$-125 + 216.5j$$

Question 7

What is the magnitude $$\rho of $$the resultant voltage $$V_4$$ = $$V_1$$ + $$V_2$$ + $$V_3 if $$the rectangular components are $$x_{total} = 410.388$$ and $$y_{total} = 178.238$$?

Accepted Answer

$$\rho = \sqrt{410.388^2$ + 178.238^2$} \approx 447.4$$

Question 8

Which of the following best describes the effect of increasing the amplitude $$A in$$ $$V_1 = A$$\sin(0.5t)$$ on the graph of the function?

Accepted Answer

The peaks and troughs of the wave become higher and lower, but the period remains unchanged.

Question 9

If $$V_3 = 36$$\sin(0.5t + C)$$ and C$ is found using $$\cos C = \frac{10.75}{36\sin(2.5)}$$, what is the value of C$ (rounded to 3 decimal places)?

Accepted Answer

$$C = 1.048$$

Question 10

A function generator produces $$V = 2\sin(2\alpha t) - 2.5\cos(\alpha t) - 2.5. If$$ $$\alpha = 43$$, which trigonometric substitution is used to convert $$\sin(2\alpha t)$$ into a function of $$\cos(\alpha t)$$?

Accepted Answer

$$\sin(2	heta) = 2\sin 	heta \cos 	heta$$

Question 11

In the context of animatronic motion, why is it important to use sinusoidal waveforms for actuator control?

Accepted Answer

Sinusoidal waveforms provide smooth, continuous motion that mimics natural movement.

Question 12

If $$V_1 = 17.96$$\sin(0.5t)$$ and V_2$ = 31.19\cos(0.5t)$$, what is the maximum possible value of $$V_3$$ = $$V_1$$ + $$V_2$$?

Accepted Answer

$$\sqrt{17.96^2$ + 31.19^2$} \approx 36$$

Question 13

A rollercoaster camera system uses a periodic voltage to trigger a flash. If the period of the voltage is reduced, what is the effect on the number of flashes per second?

Accepted Answer

The number of flashes per second increases.

Question 14

Given $$V_3 = 36$$\sin(0.5t + C)$$, which of the following is the correct expanded form using the sine addition formula?

Accepted Answer

$$36\sin(0.5t)\cos C + 36\cos(0.5t)\sin C$$

Question 15

If the resultant voltage $$V_4$$ has rectangular components $$x_{total} = 410.388$$ and $$y_{total} = 178.238$$, what is the phase angle $$	heta (in $$radians) of $$V_4$$?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{178.238}{410.388}ight)$$

Algebra & Trigonometry Study Guidance: Sinusoidal Waves, Trigonometric Identities, and Vector Functions

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