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\(t = \frac {\pi }{2} + \pi k\) for any integer \(k\).
\(\sin (t) = -\frac {\sqrt {2}}{2}\)
\(t = \frac {5\pi }{4} + 2\pi k\) or \(t = \frac {7\pi }{4} + 2\pi k\) for any integer \(k\).
\(\cos (t) = 3\)
Since \(-1 \leq \cos (t) \leq 1\), \(\cos (t) = 3\) has no (real) solution.
\(\sin (t) = -\frac {1}{2}\)
\(t = \frac {7\pi }{6} + 2\pi k\) or \(t = \frac {11\pi }{6} + 2\pi k\) for any integer \(k\).
\(\cos (t) = \frac {1}{2}\)
\(t = \frac {\pi }{3} + 2\pi k\) or \(t = \frac {5\pi }{3} + 2\pi k\) for any integer \(k\).
\(\sin (t) = -2\)
Since \(-1 \leq \sin (t) \leq 1\), \(\sin (t) = -2\) has no (real) solution.
\(\cos (t) = 1\)
\(t = 2\pi k\) for any integer \(k\).
\(\sin (t) = 1\)
\(t = \frac {\pi }{2} + 2\pi k\) for any integer \(k\).
\(\cos (t) = -\frac {\sqrt {2}}{2}\)
\(t = \frac {3\pi }{4} + 2\pi k\) or \(t = \frac {5\pi }{4} + 2\pi k\) for any integer \(k\).
In Exercises pointsfirst - pointslast, let \(\theta \) be the angle in standard position whose terminal side contains the given point then compute \(\cos (\theta )\) and
\(\sin (\theta )\).
In Exercises calculatorfirst - calculatorlast, use your calculator to approximate the given value to three decimal places. Make sure your calculator is in the
proper angle measurement mode!
For \(f(t) = 3t + \sin (2t)\), find functions \(g\) and \(h\) so that \(f=g+h\).
One solution is \(g(t) = 3t\) and \(h(t) = \sin (2t)\).
For \(f(\theta ) = 3\cos (\theta ) - \sin (4\theta )\), find functions \(g\) and \(h\) so that \(f=g-h\).
One solution is \(g(\theta ) = 3 \cos (\theta )\) and \(h(\theta ) = \sin (4 \theta )\).
For \(f(t) = e^{-0.1t} \sin (3t)\), find functions \(g\) and \(h\) so that \(f=gh\).
One solution is \(g(t) = e^{-0.1t}\) and \(h(t) = \sin (3t)\).
For \(r(t) = \frac {\sin (t)}{t}\), find functions \(f\) and \(g\) so \(r = \frac {f}{g}\).
One solution is \(f(t) = \sin (t)\) and \(g(t) = t\).
For \(r(\theta ) =\sqrt {3 \cos (\theta )}\), find functions \(f\) and \(g\) so \(r = g \circ f\).
One solution is \(f(\theta ) = 3 \cos (\theta )\) and \(g(\theta ) = \sqrt {\theta }\).
For each function \(S(t)\) listed below, compute the average rate of change over the indicated interval. (See Definition arc in
Section AverageRateofChange for a review of this concept, as needed.) What trends do you notice? Be sure your calculator is in radian
mode!
In Exercises motionfirst - motionlast, find the equations of motion for the given scenario. Assume that the center of the motion is the origin, the
motion is counter-clockwise and that \(t = 0\) corresponds to a position along the positive \(x\)-axis. (See Equation equationsforcircularmotion and Example
EarthRotationEx.)
Recall: The diameter is 128 feet and completes 2 revolutions in 2 minutes, 7 seconds.
\(r = 64\) feet, \(\omega = \frac {4\pi }{127} \, \frac {\text {radians}}{\text {second}}\), \(x = 64 \cos \left (\frac {4\pi }{127} \, t \right )\), \(y = 64 \sin \left (\frac {4\pi }{127} \, t \right )\). Here \(x\) and \(y\) are measured in feet and \(t\) is measured in seconds
Consider the numbers: \(0\), \(1\), \(2\), \(3\), \(4\). Take the square root of each of these numbers, then divide each by \(2\). The resulting numbers
should look hauntingly familiar.
On page ??, we see that the sine and cosine functions of angles can be considered functions of real numbers. With help
from your classmates, discuss the domains and ranges of \(f(t) = \sin (t)\) and \(g(t) = \cos (t)\). Write your answers using interval notation.
Another way to establish Theorem cosinesinecircle is to use transformations. Re-read the discussion following Theorem standardcirclealternate in Chapter TheConicSections and
transform the Unit Circle, \(x^2+y^2 = 1\), to \(x^2 + y^2 = r^2\) using horizontal and vertical stretches. Show if the coordinates on the Unit Circle are \((\cos (\theta ), \sin (\theta ))\), then the
corresponding coordinates on \(x^2+y^2 = r^2\) are \((r \cos (\theta ), r \sin (\theta ))\).
In the scenario of Equation equationsforcircularmotion, we assumed that at \(t=0\), the object was at the point \((r,0)\). If this is not the case, we can adjust the
equations of motion by introducing a ‘time delay.’ If \(t_{0} > 0\) is the first time the object passes through the point \((r,0)\), show, with the help of
your classmates, the equations of motion are \(x = r \cos (\omega (t - t_{0}))\) and \(y = r \sin (\omega (t-t_{0}))\).
