# Welcome to Calculus 1000A 002!

## Class info

• Instructor name: Pál Zsámboki
• Instructor email: pzsambok@uwo.ca
• Instructor office: MC 133
• Class webpage: on OWL
• Documents on OWL page:
• Class Outline written by the Course Coordinator
• List of practice problems
• Textbook: Single Variable Calculus: Early Transcendentals (8th edition), by James Stewart (Brooks/Cole)
• It is important that you have access to the correct edition, since there is a list of practice problems we'll suppose you know how to solve.
• Office hours: Thursdays 3pm-5pm. No office hour on September 23.
• Math Help Centre: TBD, starting on September 19

• Midterm exam 35%
• The midterm exam will be on Friday October 21, from 7pm-9pm. Location TBA. The exam will be closed book, no notes, no calculators or any other electronic devices will be permitted.
• Final exam 50%
• The final exam will be cumultative, and it will be 3 hours long. Location TBA. The exam will be closed book, no notes, no calculators or any other electronic devices will be permitted.
• Quizzes 15%
• There will be 6 online quizzes on OWL, with 2 questions in each.
• You will have 2 weeks for every one of them, and you'll have 3 tries with each.
• Each quiz will have a time limit of 20 minutes.
• For more details, check quizzes.txt.

# Trigonometry

## Parametrization of the unit circle

• I think the most intuitive way of thinking about trigonometric functions is that the function $$f(\theta)=(\cos(\theta),\sin(\theta))$$ is the parametrization of the unit circle with respect to arc length, in counterclockwise or in other words positive direction, starting from the point $$f(0)=(1,0)$$.
• That is, the function $$f(\theta)$$ traces out the unit circle as $$\theta$$ varies
• Being parametrized with respect to arc length means that if we change $$\theta$$ by 1 unit, the parametrization $$f(\theta)$$ traces out an arc of length 1.

## Standard position, positive and negative angles

• The standard parametrization $$f(\theta)=(\cos(\theta),\sin(\theta))$$ corresponds to the following conventions.
• An angle is said to be in standard position, if its vertex is placed at the origin of the coordinate system we're using, and its initial side is on the positive $$x$$-axis
• A positive angle is obtained if going from the initial angle to the terminal angle is a counterclockwise rotation
• A negative angle is obtained if going from the initial angle to the terminal angle is a clockwise rotation

## Circles and arcs of general radii

• If the angle $$\theta$$ is measured like this, that is in the way that the parametrization of the unit circle $$f(\theta)=(\cos(\theta),\sin(\theta))$$ is with respect to arc length, we say that it is measured in radians. Since the circumference of the unit circle is $$2\pi$$, a complete revolution is done by changing $$\theta$$ by $$2\pi$$.
• Let $$r>0$$ be a positive number. Then similarly we can parametrize the circle $$C$$ with centre the origin and radius $$r$$ via $$f(\theta)=(r\cos(\theta),r\sin(\theta))$$.
• Note that this parametrization is not with respect to arc length: a complete revolution around $$C$$, which traces out a path of length the circumference of $$C$$, that is $$a=2\pi r$$, corresponds to a change of $$2\pi$$ in $$\theta$$.
• More generally, for a circular arc of length $$a$$, radius $$r$$, and angle $$\theta$$, we get the formula $$a=\theta r$$.

## Degrees

• There is another, historical angle unit, that of degrees. If angles are measured in degrees, then a complete revolution is $$360^\circ$$.
• This gives us the conversion formula between radians and degrees: $$2\pi=360^\circ$$.
• A related unit of length which I'm rather fond of is that of nautical miles (NM). 1 nautical mile is 1 latidual degree minute. That is, if you travel 60 NM from North to South, then you travelled precisely $$\tfrac{1}{360}$$ of the latitudal circumference of the Earth.
• Exercises: D.2,4,8,10

## Trigonometric functions

• Consider a right triangle $$T$$, where one of the non-right angles is $$\theta$$.
• Suppose that the angle $$\theta$$ is in standard position, let's give it the positive orientation, and let's denote the length of the hypotenuse by $$r$$.
• Then we can study the triangle $$T$$ with the parametrization $$f(\theta)=(x=r\cos(\theta),y=r\sin(\theta))$$.
• Its vertices are going to be $$O(0,0)$$, $$X(x,0)$$, and $$P(x,y)$$.
• The side $$\overline{OX}$$ is the adjacent, the side $$\overline{XP}$$ is the opposite, and the side $$\overline{OP}$$ is the hypotenuse.
• The lengths of the legs are $$|OX|=x$$, and $$|XP|=y$$.
• Then we can express the trigonometric functions using these side lengths:
• $$\cos(\theta)=\frac{x}{r}$$, $$\sin(\theta)=\frac{y}{r}$$, $$\tan(\theta)=\frac{y}{x}=\frac{\sin(\theta)}{\cos(\theta)}$$,
• $$\sec(\theta)=\frac{r}{x}=\frac{1}{\cos(\theta)}$$, $$\csc(\theta)=\frac{r}{y}=\frac{1}{\sin(\theta)}$$, $$\cot(\theta)=\frac{x}{y}=\frac{\cos(\theta)}{\sin(\theta)}$$
• Since the parametrization $$f(\theta)$$ is defined for all values of $$\theta$$, we can use the same formulas in case $$\theta<0$$ or $$\theta\ge\frac{\pi}{2}$$.

## Graphs of the trigonometric functions

• Note the following properties on the graphs of $$\cos(x)$$ and $$\sin(x)$$.
• Both functions are $$2\pi$$-periodic, their domain is $$(-\infty,\infty)$$, and their range is $$[-1,1]$$.
• $$\cos(x)=\sin(x+\frac{\pi}{2})$$. This can be also seen by rotating the right triangle $$OXP$$ by $$\frac{\pi}{2}$$.
• $$\cos(x)=0$$ iff (if and only if) $$x=\frac{\pi}{2}+k\pi$$ for some integer $$k$$, and $$\sin(x)=0$$ iff $$x=k\pi$$.
• I think it makes remembering the special values easier to notice that on the interval $$[0,\frac{\pi}{2}]$$, the functions $$\cos(x)$$ is decreasing, and the function $$\sin(x)$$ is increasing.