Mathematics 265 Introduction to Calculus I

Study Guide :: Unit 1

Brief Review of Algebra and Trigonometry for Calculus

Trigonometry

Definitions of Trigonometric Functions

Consider the unit circle (centred at the origin ${\displaystyle 0}$ and with radius ${\displaystyle 1}$) on the Cartesian plane. Each point ${\displaystyle P}$ on this circle has coordinates $(x_{\theta },y_{\theta })$, where the ${\displaystyle \theta }$ refers to the angle from the point $(1,0)$ to $P$. The angle ${\displaystyle \theta }$ is a positive angle if it is measured in the counterclockwise direction; it is negative if it is measured in the clockwise direction, as shown in Figure 1.1, below.

Figure 1.1. Direction of movement for angle ${\displaystyle \theta }$

For each angle ${\displaystyle \theta }$, consider the corresponding point $P=(x_{\theta },y_{\theta })$ on the unit circle, and define:

These trigonometric functions are defined for angles, not real numbers. In calculus, however, we work with these functions defined on real numbers; that is, in radians. To do so, we define a “winding function”—a correspondence between real numbers and angles—as follows.

Consider again the unit circle with centre ${\displaystyle 0}$, and let ${\displaystyle P_0}$ be the point ${\displaystyle \left(1,0\right)}$. For each number ${\displaystyle a\text{,}}$ we associate a point ${\displaystyle P_a}$ on the unit circle using the rules given below.

• If ${\displaystyle a>0}$, then we start at ${\displaystyle P_0}$ and move around the circle in the counterclockwise direction, until we have traced out a path whose length is ${\displaystyle a}$. The point where our path ends is ${\displaystyle P_a}$. See Figure 1.2, below.

  

  Figure 1.2. ${\displaystyle P_a\text{,}a>0}$

• If ${\displaystyle a<0}$, then we start at ${\displaystyle P_0}$ and move around the circle in the clockwise direction, until we have traced out a path whose length is ${\displaystyle \left|a\right|}$. The point where our path ends is ${\displaystyle P_a}$. See Figure 1.3, below.

  

  Figure 1.3. ${\displaystyle P_a\text{,}a<0}$

In either case, we associate the angle ${\displaystyle \angle P_{0}0P_a}$ to this number ${\displaystyle a}$.

Remarks 1.1

1. The circumference of the unit circle is ${\displaystyle 2\pi }$, so, to this number, we associate the angle ${\displaystyle 360°}$. That is, the angle that corresponds to ${\displaystyle \pi }$ (radians) is ${\displaystyle 180°}$.
2. The numbers ${\displaystyle a}$ and ${\displaystyle a+2\pi }$ correspond to the same angles, since ${\displaystyle P_a}$ and ${\displaystyle P_{a+2\pi }}$ coincide on the unit circle. The same is true for ${\displaystyle a+2k\pi }$, for any integer ${\displaystyle k}$.
3. When we define a trigonometric function on a real number ${\displaystyle a}$, we consider this association; that is, ${\displaystyle \text{sin }a=\text{sin }\angle P_{0}0P_a}$. For example, ${\displaystyle \text{sin }\pi =\text{sin }180°}$.
4. We always work with numbers (radians) in this course.

As you can see from this discussion, the coordinates of the point ${\displaystyle P}$ are ${\displaystyle \left(\text{cos }\theta \text{,}\text{ sin }\theta \right)}$. Furthermore, since ${\displaystyle \theta =\theta +2k\pi }$ for any integer ${\displaystyle k}$, to show a trigonometric identity or property for any number ${\displaystyle \theta }$, it is enough to show it for the corresponding number between 0 and ${\displaystyle 2\pi }$.

Therefore,

• If ${\displaystyle \theta =0}$, the coordinates of ${\displaystyle P}$ are ${\displaystyle \left(1\text{,}0\right)}$; hence,

  ${\displaystyle \text{cos }0=1\text{and}\text{sin }0=0}$.

• If ${\displaystyle \theta =\frac{\pi }{2}}$, the coordinates of ${\displaystyle P}$ are ${\displaystyle \left(0\text{,}1\right)}$; hence,

  ${\displaystyle \text{cos }\frac{\pi }{2}=0\text{and}\text{sin }\frac{\pi }{2}=1}$.

• If ${\displaystyle \theta =\pi }$, the coordinates of ${\displaystyle P}$ are ${\displaystyle \left(-1\text{,}0\right)}$; hence,

  ${\displaystyle \text{cos }\pi =-1\text{and}\text{sin }\pi =0}$.

• If ${\displaystyle \theta =\frac{3}{2}\pi }$, the coordinates of ${\displaystyle P}$ are ${\displaystyle \left(0\text{,}-1\right)}$; hence,

  ${\displaystyle \text{cos }\frac{3}{2}\pi =0\text{and}\text{sin }\frac{3}{2}\pi =-1}$.

Consider the right-angle (rectangular) triangle ${\displaystyle T}$ in Figure 1.4, below, with vertices ${\displaystyle ABC}$, legs ${\displaystyle a}$ and ${\displaystyle b}$, and hypotenuse ${\displaystyle c>1}$. Then, ${\displaystyle \theta =BAC}$ is the angle at the vertex ${\displaystyle A}$.

Figure 1.4. Right-angle triangle, ${\displaystyle T}$, with a hypotenuse ${\displaystyle c>1}$

Next, consider the rectangular triangle ${\displaystyle R}$ with sides ${\displaystyle x\text{,}y\text{,}z}$, hypotenuse ${\displaystyle z=1}$ and the angle at the vertex ${\displaystyle Y}$ equal to ${\displaystyle \theta =XYZ}$, as shown in Figure 1.5, below. The coordinates of the vertex ${\displaystyle Y}$ are ${\displaystyle \left(\text{cos }\theta \text{,}\text{ sin }\theta \right)}$; that is, ${\displaystyle x=\text{cos }\theta }$ and ${\displaystyle y=\text{sin }\theta }$.

Figure 1.5. Right-angle triangle, ${\displaystyle R}$, with a hypotenuse ${\displaystyle z=1}$

The triangles ${\displaystyle T}$ (Figure 1.4) and ${\displaystyle R}$ (Figure 1.5) are similar. [Why?]

So,

${\displaystyle \frac{a}{c}=\frac{y}{z}=\frac{\text{sin }\theta }{1}\text{and}\frac{b}{c}=\frac{x}{z}=\frac{\text{cos }\theta }{1}\text{.}}$

We conclude that

${\displaystyle \text{sin }\theta =\frac{a}{c}\text{and}\text{cos }\theta =\frac{b}{c}\text{.}}$

Therefore,

${\displaystyle \text{tan }\theta =\frac{a}{b}\text{;}\text{cot }\theta =\frac{b}{a}\text{;}\text{sec }\theta =\frac{c}{b}\text{;}\text{csc }\theta =\frac{c}{a}\text{.}}$

You should convince yourself that the same argument can be used to explain why the values of the trigonometric functions are the same, even if the triangle ${\displaystyle T}$ has hypotenuse smaller than 1; that is, when the triangle is as shown in Figure 1.6, below.

Figure 1.6. Right-angle triangle, ${\displaystyle T}$, with a hypotenuse ${\displaystyle c<1}$

Exact Values of the Trigonometric Functions

See the table of exact values for sine and cosine on page 358 of the textbook.

The exact values for ${\displaystyle \theta =45°}$ can be deduced from the right-angle triangle shown in Figure 1.7, below. The acute angles are ${\displaystyle 45°}$ each; hence,

${\displaystyle \text{sin }45°=\text{sin }\frac{\pi }{4}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\text{and}\text{cos }45°=\text{cos }\frac{\pi }{4}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\text{.}}$

Figure 1.7. Right-angle triangle, with acute angles of ${\displaystyle 45°}$

The exact values for ${\displaystyle 60°}$ and ${\displaystyle 30°}$ are deduced from the right-angle triangle made up of half of an equilateral triangle of side ${\displaystyle 2}$ (see Figure 1.8), below.

Figure 1.8. Right-angle triangle, half of an equilateral triangle of side 2

The height of this triangle is ${\displaystyle \sqrt{3}}$ and the acute angles are ${\displaystyle 30°=\pi ∕6}$ and ${\displaystyle 60°=\pi ∕3}$; hence,

${\displaystyle \begin{array}{cc}\text{sin }60°=\text{sin }{\displaystyle \frac{\pi }{3}}={\displaystyle \frac{\sqrt{\text{3}}}{\text{2}}}& \text{cos }60°=\text{cos }{\displaystyle \frac{\pi }{\text{3}}}={\displaystyle \frac{1}{\text{2}}}\\ \text{sin }30°=\text{sin }{\displaystyle \frac{\pi }{6}}={\displaystyle \frac{1}{\text{2}}}& \text{cos }30°=\text{cos }{\displaystyle \frac{\pi }{\text{6}}}={\displaystyle \frac{\sqrt{\text{3}}}{\text{2}}}\end{array}}$

Study Figure 1.9, below.

Figure 1.9. Point P at ${\displaystyle \left(\text{cos }\theta \text{,}\text{ sin }\theta \right)}$

If the angle ${\displaystyle \theta =\frac{2}{3}\pi }$, then ${\displaystyle \alpha =\frac{1}{3}\pi }$.

The coordinates of ${\displaystyle P}$ are ${\displaystyle \left(\text{cos }\theta \text{,}\text{ sin }\theta \right)}$. On the other hand, from the rectangular triangle, we see that the legs of ${\displaystyle P}$ are ${\displaystyle \text{cos }\alpha }$ (horizontal) and ${\displaystyle \text{sin }\alpha }$ (vertical). The first coordinate of ${\displaystyle P}$ is negative and the second coordinate is positive. Therefore, ${\displaystyle \text{cos }\theta =-\text{cos }\alpha }$, and ${\displaystyle \text{sin }\theta =\text{sin }\alpha }$, and

${\displaystyle \text{cos }\frac{2}{3}\pi =-\frac{1}{2}\text{,}\text{and}\text{sin }\frac{2}{3}\pi =\frac{\sqrt{3}}{2}\text{.}}$