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# Week 4 Problem Set C.
Due: Friday 9/29
I know I know, more homework. But doing is learning! We review some geometry and trigonometry below.
### Reading.
Finish reading either (or both):
- Thomas' Calculus (12E) chapters 1.1, 1.2, and 1.3
- Stewart's Calculus (9E, with Clegg and Watson) chapters 1.1, 1.2, and 1.3
### Warm-up geometry: More square.
Find the area of the following square, again.
![[1 teaching/smc-fall-2023-math-7/week-4/---files/week-4C-problems 2023-09-21 10.36.36.excalidraw.svg]]
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### Radians and degrees.
1. A central angle in a circle of radius $8$ is subtended by an arc of length $10\pi$. Find the angle's radian and degree measures.
2. If you roll a 1 meter diameter wheel forward 30 cm over level ground, through what angle will the wheel turn? Answer it in radians and then in degrees as well. You can approximate your answer to one decimal place.
### Evaluating trigonometric functions.
1. Copy and complete the following table of function values. If the function is undefined at a given angle, write "undefined". Give exact values, and do note use a calculator. $$
\begin{array}{c|ccccc}
\theta & -\pi & -2\pi / 3 & 0 & \pi / 2 & 3\pi / 4 \\ \hline
\sin\theta \\
\cos\theta \\
\tan\theta \\
\cot\theta \\
\sec\theta \\
\csc\theta
\end{array}
$$
2. Copy and complete the following table of function values. If the function is undefined at a given angle, write "undefined". Give exact values, and do not use a calculator. $$
\begin{array}{c|ccccc}
\theta & -3\pi / 2 & -\pi / 3 & -\pi / 6 & \pi / 4 & 5\pi / 6 \\ \hline
\sin\theta \\
\cos\theta \\
\tan\theta \\
\cot\theta \\
\sec\theta \\
\csc\theta
\end{array}
$$
3. In each of the following, one of $\sin(x),\cos(x),$ or $\tan(x)$ is to you. Find the **other two** if $x$ is in the specified interval.
1. $\sin(x)= \frac{3}{5}$, with $x\in [\frac{\pi}{2},\pi]$
2. $\tan(x)=2$, with $x\in[0,\frac{\pi}{2}]$
3. $\cos(x)=\frac{1}{3}$, with $x\in [-\frac{\pi}{2},0]$
4. $\cos(x)=- \frac{5}{13}$, with $x\in[\frac{\pi}{2},\pi]$
5. $\tan(x)=\frac{1}{2}$, with $x\in[\pi,\frac{3\pi}{2}]$
6. $\sin(x)=-\frac{1}{2}$, with $x\in[\pi,\frac{3\pi}{2}]$
4. Give a sketch of the following function below. What is the period of each function?
1. $\sin(2x)$
2. $\sin(x /2)$
3. $\cos(\pi x /2)$
4. $\sin(x + \frac{\pi}{6})$
5. $\cos(x+ \frac{2\pi}{3})-2$
### Using the addition formulas.
1. Use the addition formulas to derive the following identities:
1. $\sin(x- \frac{\pi}{2}) = -\cos(x)$
2. $\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)$
3. $\sin(A-B)=\sin(A)\cos(B)-\cos(A)\sin(B)$
4. $\displaystyle \cos(A)\cos(B) = \frac{1}{2}[\cos(A+B)+\cos(A-B)]$ , this is the prosthaphaeresis identity.
2. Evaluate $\sin(\frac{7\pi}{12})$ by noting it is the same as $\sin(\frac{\pi}{4}+\frac{\pi}{3})$.
3. Evaluate $\cos(\frac{11\pi}{12})$ by noting it is the same as $\cos(\frac{\pi}{4}+\frac{2\pi}{3})$
4. Evaluate $\cos(\frac{\pi}{12})$
5. Evaluate $\sin(\frac{5\pi}{12})$
### Using double-angle and half-angle formulas.
Find the function values of the following:
1. $\cos^{2}(\frac{\pi}{8})$
2. $\cos^{2}(\frac{5\pi}{12})$
3. $\sin^{2}(\frac{\pi}{12})$
4. $\sin^{2}(\frac{3\pi}{8})$
### Theory and practice.
1. **The tangent sum formula.** The standard formula for the tangent of the sum of two angles is $$
\tan(A+B) = \frac{\tan A + \tan B}{1-\tan A \tan B}
$$ Derive this formula.
2. Use the law of cosine to the following triangle in a unit circle (radius 1) in the diagram to derive the formula for $\cos(A-B)$:![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230921112436.png]] (Hint: Write out the coordinates of the two vertices of the triangle on the circle, and find their distance.)
3. A triangle with sides $a=2$ and $b=3$ has an angle between them $C=60^{\circ}$. Find the third side length $c$.
4. A triangle with sides $a=2$ and $b=3$ has an angle between them $C=40^{\circ}$. Find the third side length $c$.
### Approximation of $\sin(x)$.
In the textbook (end of section 1.3 of Thomas) and in class (hopefully soon), we see that for any angle $\theta$, we always have these inequalities: $$
-|\theta| \le \sin \theta \le|\theta|\quad\text{and}\quad -|\theta|\le 1-\cos\theta \le|\theta|
$$
There is another approximation that is good to know: When $\theta$ is really, really, really small (close to zero), then $$
\sin\theta \approx \theta, \quad\text{when }|\theta|\ll 1.
$$
This is an important and useful approximation called **small angle approximation**, and is often used in physics and engineering (without you noticing sometimes!)
Let us just numerically verify that this is the case. Using a calculator, calculate the following (do it in **radian**!)
$$
\begin{array}{}
\theta & \sin(\theta) & \text{relative error} \\\hline
1 \\
0.1 \\
0.01 \\
0.001 \\
0.0001
\end{array}
$$where relative error is the how far off is $\theta$ as an approximation to $\sin\theta$, as compared to the true value of $\sin\theta$, so compute $$
\text{relative error} = \frac{|\theta - \sin\theta|}{|\sin\theta|}
$$
What can you say about the relative error as $\theta$ gets closer and closer to $0$?
Later we will see that by examining what happens to the relative error as $\theta\to0$ is the idea of a **limit**, which we will explore next week.
//// Excellent work ! ////