ihatemath0101
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Calculus help: properties of the trigonometric functions, range, maximum height, etc
(2) Pretend that Angry Birds are real. The path of an Angry Bird fired at an inclination \(\displaystyle \theta\) to the horizontal with initial speed \(\displaystyle v_0\) is a parabola. The range of the Angry Bird (the horizontal distance that it travels) is given by:
. . .\(\displaystyle R\left(\theta,\, v_0\right)\, =\, \dfrac{v_0^2\, \sin\left(2\theta\right)}{g}\)
...where \(\displaystyle g\, \approx\, 9.8\, m/s^2\) is the acceleration due to gravity. The maximum height attained by the Angry Bird is given by:
. . .\(\displaystyle H\left(\theta,\, v_0\right)\, =\,\dfrac{v_0^2\, \sin^2\left(\theta\right)}{2g}\)
Find the range and maximum height of an Angry Bird fired at an angle 30° with an initial speed 200 m/s.
(3) Use properties of the trigonometric functions to find (without a calculator) the exact value of:
. . .\(\displaystyle \sin\left(-\dfrac{\pi}{12}\right)\, \csc\left(\dfrac{25\pi}{12}\right)\)
(4) The voltage \(\displaystyle V\), in volts, produced by an AC generator at time \(\displaystyle t\), in seconds, is \(\displaystyle V(t)\, =\, 220\, \sin(120\pi t).\)
(a) Graph \(\displaystyle V\) over two periods, beginning at \(\displaystyle t\, =\, 0\).
(b) If a resistance of \(\displaystyle R\, =\, 10\) ohms is present, what is the current \(\displaystyle I\)? (Use Ohm's Law: \(\displaystyle V\, =\, IR\))
(c) What is the amplitude and period of the current \(\displaystyle I\)?
(d) Graph \(\displaystyle I\) over two periods, beginning at \(\displaystyle t\, =\, 0\).
(2) Pretend that Angry Birds are real. The path of an Angry Bird fired at an inclination \(\displaystyle \theta\) to the horizontal with initial speed \(\displaystyle v_0\) is a parabola. The range of the Angry Bird (the horizontal distance that it travels) is given by:
. . .\(\displaystyle R\left(\theta,\, v_0\right)\, =\, \dfrac{v_0^2\, \sin\left(2\theta\right)}{g}\)
...where \(\displaystyle g\, \approx\, 9.8\, m/s^2\) is the acceleration due to gravity. The maximum height attained by the Angry Bird is given by:
. . .\(\displaystyle H\left(\theta,\, v_0\right)\, =\,\dfrac{v_0^2\, \sin^2\left(\theta\right)}{2g}\)
Find the range and maximum height of an Angry Bird fired at an angle 30° with an initial speed 200 m/s.
(3) Use properties of the trigonometric functions to find (without a calculator) the exact value of:
. . .\(\displaystyle \sin\left(-\dfrac{\pi}{12}\right)\, \csc\left(\dfrac{25\pi}{12}\right)\)
(4) The voltage \(\displaystyle V\), in volts, produced by an AC generator at time \(\displaystyle t\), in seconds, is \(\displaystyle V(t)\, =\, 220\, \sin(120\pi t).\)
(a) Graph \(\displaystyle V\) over two periods, beginning at \(\displaystyle t\, =\, 0\).
(b) If a resistance of \(\displaystyle R\, =\, 10\) ohms is present, what is the current \(\displaystyle I\)? (Use Ohm's Law: \(\displaystyle V\, =\, IR\))
(c) What is the amplitude and period of the current \(\displaystyle I\)?
(d) Graph \(\displaystyle I\) over two periods, beginning at \(\displaystyle t\, =\, 0\).
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