14.5: Exercises

14.1. Radians, degrees and right triangles.

Convert the following expressions in radians to degrees:

(a) \(\pi\)

(b) \(5 \pi / 3\)

(c) \(21 \pi / 23\)

(d) \(24 \pi\)

Convert the following expressions in degrees to radians:

(e) \(100^{\circ}\)

(f) \(8^{o}\)

(g) \(450^{\circ}\)

(h) \(90^{\circ}\)

Using a right triangle, evaluate each of the following:

(i) \(\cos (\pi / 3)\)

(j) \(\sin (\pi / 4)\)

(k) \(\tan (\pi / 6)\)

14.2. Graphing functions. Graph the following functions over the indicated ranges:

(a) \(y=x \sin (x)\) for \(-2 \pi<x<2 \pi\)

(b) \(y=e^{x} \cos (x)\) for \(0<x<4 \pi\)

14.3. Sketching the graph. Sketch the graph for each of the following functions:

(a) \(y=\frac{1}{2} \sin 3\left(x-\frac{\pi}{4}\right)\)

(b) \(y=2-\sin x\)

(c) \(y=3 \cos 2 x\)

(d) \(y=2 \cos \left(\frac{1}{2} x+\frac{\pi}{4}\right)\)

14.4. Converting angles. The radian is an important unit associated with angles. One revolution about a circle is equivalent to 360 degrees or \(2 \pi\) radians.

(a) Convert the following angles (in degrees) to angles in radians. Express each as multiples of \(\pi\), not as decimal expansions:

(i) 45 degrees (ii) 30 degrees (iii) 60 degrees (iv) 270 degrees.

(b) Find the sine and the cosine of each of the above angles.

14.5. Trigonometric functions and rhythmic functions. Find the appropriate trigonometric function to describe the following rhythmic processes:

(a) Daily variations in the body temperature \(T(t)\) of an individual over a single day, with the maximum of \(37.5^{\circ} \mathrm{C}\) at \(8: 00\) am and a minimum of \(36.7^{\circ} \mathrm{C} 12\) hours later.

(b) Sleep-wake cycles with peak wakefulness \((W=1)\) at 8:00 am and 8:00pm and peak sleepiness \((W=0)\) at \(2: 00 \mathrm{pm}\) and \(2: 00\) am.

In both cases, express \(t\) as time in hours with \(t=0\) taken at \(0: 00 \mathrm{am}\).

14.6. Trigonometric functions and rhythmic functions. Find the appropriate trigonometric function to describe the following rhythmic processes:

(a) The displacement \(S \mathrm{~cm}\) of a block on a spring from its equilibrium position, with a maximum displacement \(3 \mathrm{~cm}\) and minimum displacement \(-3 \mathrm{~cm}\), a period of \(\frac{2 \pi}{\sqrt{g / l}}\) and at \(t=0, S=3\).

(b) The vertical displacement \(y\) of a boat that is rocking up and down on a lake, with \(y\) measured relative to the bottom of the lake. It has a maximum displacement of 12 meters and a minimum of 8 meters, a period of 3 seconds, and an initial displacement of 11 meters when measurement was first started (i.e., \(t=0\) ).

14.7. Sunspot cycles. The number of sunspots (solar storms on the sun) fluctuates with roughly 11-year cycles with a high of 120 and a low of 0 sunspots detected. A peak of 120 sunspots was detected in the year 2000 .

Which of the following trigonometric functions could be used to approximate this cycle?

(a) \(N=60+120 \sin \left(\frac{2 \pi}{11}(t-2000)+\frac{\pi}{2}\right)\)

(b) \(N=60+60 \sin \left(\frac{11}{2 \pi}(t+2000)\right)\)

(c) \(N=60+60 \cos \left(\frac{11}{2 \pi}(t+2000)\right)\)

(d) \(N=60+60 \sin \left(\frac{2 \pi}{11}(t-2000)\right)\)

(e) \(N=60+60 \cos \left(\frac{2 \pi}{11}(t-2000)\right)\)

14.8. Inverse trigonometric functions. As seen in Section 14.3, the inverse trigonometric function \(\arctan (x)\) (also written \(\tan ^{-1}(x)\) ) means the angle \(\theta\) where \(-\pi / 2<\theta<\pi / 2\) whose tan is \(x\). Thus \(\cos (\arctan (x)\) (or \(\cos \left(\tan ^{-1}(x)\right)\) is the cosine of that same angle. By using a right triangle whose sides have length \(1, x\) and \(\sqrt{1+x^{2}}\) we can verify that

\[\cos (\arctan (x))=1 / \sqrt{1+x^{2}} \nonumber \]

Use a similar geometric argument to arrive at a simplification of the following functions:

(a) \(\sin (\arcsin (x))\)

(b) \(\tan (\arcsin (x)\)

(c) \(\sin (\arccos (x)\).

14.9. Inverse trigonometric functions. The value of \(\tan (\arccos (x))\) is which of the following?

(a) \(1-x^{2}\)

(b) \(x\)

(c) \(1+x^{2}\)

(d) \(\frac{\sqrt{1-x^{2}}}{x}\)

(e) \(\frac{\sqrt{1+x^{2}}}{x}\)

14.10. Inverse trigonometric functions. The function \(y=\tan (\arctan (x))\) has which of the following for its domain and range?

(a) Domain \(0 \leq x \leq \pi ;\) Range \(-\infty \leq y \leq \infty\)

(b) Domain \(-\infty \leq x \leq \infty\); Range \(-\infty \leq y \leq \infty\)

(c) Domain \(-\pi \leq x \leq \pi\); Range \(-\pi \leq y \leq \pi\)

(d) Domain \(-\pi / 2 \leq x \leq \pi / 2\); Range \(-\pi / 2 \leq y \leq \pi / 2\)

(e) Domain \(-\infty \leq x \leq \infty\); Range \(0 \leq y \leq \pi\)

14.11. Simplify trigonometric identity.

(a) Use a double-angle trigonometric identity to simplify the following expression as much as possible:

\[y=\cos (2 \arcsin (x)) \nonumber \]

(b) For what values of \(x\) is this simplification possible?