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7.1: Trigonometry

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In this chapter we shall study the trigonometric functions, i.e., the sine and cosine function and other functions that are built up from them. Let us start from the beginning and introduce the basic concepts of trigonometry.

The unit circle x2+y2=1 has radius 1 and center at the origin.

Two points P and Q on the unit circle determine an arc PQ, an angle POQ, and a sector POQ. The arc starts at P and goes counterclockwise to Q along the circle. The sector POQ is the region bounded by the arc PQ and the lines OP and OQ. As Figure 7.1.1 shows, the arcs PQ and QP are different.

2024_06_12_6bbb2ed052b26fd5e777g-01.jpg
Figure 7.1.1

Trigonometry is based on the notion of the length of an arc. Lengths of curves were introduced in Section 6.3. Although that section provides a useful background, this chapter can also be studied independently of Chapter 6. As a starting point we shall give a formula for the length of an arc in terms of the area of a sector. (This formula was proved as a theorem in Section 6.3 but can also be taken as the definition of arc length.)

Definition

The length of an arc PQ on the unit circle is equal to twice the area of the sector POQ,s=2A.

This formula can be seen intuitively as follows. Consider a small arc PQ of length s (Figure 7.1.2). The sector POQ is a thin wedge which is almost a right triangle of altitude one and base Δs. Thus ΔA12Δs. Making Δs infinitesimal and adding up, we get A=12s.

The number π3.14159 is defined as the area of the unit circle. Thus the unit circle has circumference 2π.

The area of a sector POQ is a definite integral. For example, if P is the point P(1,0) and the point Q(x,y) is in the first quadrant, then we see from Figure 7.1.3 that the area is

A(x)=12x1x2+1x1t2dt

Notice that A(x) is a continuous function of x. The length of an are has the following basic property.

2024_06_12_6bbb2ed052b26fd5e777g-02(1).jpg
Figure 7.1.2
2024_06_12_6bbb2ed052b26fd5e777g-02.jpg
Figure 7.1.3
Theorem 1

Let P be the point P(1,0). For every number s between 0 and 2π there is a point Q on the unit circle such that the are PQ has length s.

PROOF

We give the proof for s between 0 and π/2, whence

012sπ/4

Let A(x) be the area of the sector POQ where Q=Q(x,y) (Figure 7.1.4). Then A(0)=π/4,A(1)=0 and the function A(x) is continuous for 0x1. By the Intermediate Value Theorem there is a point x0 between 0 and 1 where the sector has area 12s,

A(x0)=12s

Therefore the arc PQ has length

2A(x0)=s

2024_06_12_6bbb2ed052b26fd5e777g-03(1).jpg
Figure 7.1.4: Copy and Paste Caption here. (Copyright; author via source)

Arc lengths are used to measure angles. Two units of measurement for angles are radians (best for mathematics) and degrees (used in everyday life).

Definition

Let P and Q be two points on the unit circle. The measure of the angle POQ in radians is the length of the arc PQ. A degree is defined as

1=π/180 radians 

whence the measure of POQ in degrees is 180/π times the length of PQ.

Approximately, 1^{\circ} \sim 0.01745 radians,

1 \text { radian } \sim 57^{\circ} 18^{\prime}=\left(57 \frac{18}{60}\right)^{\circ} \nonumber

A complete revolution is 360^{\circ} or 2 \pi radians. A straight angle is 180^{\circ} or \pi radians. A right angle is 90^{\circ} or \pi / 2 radians.

It is convenient to take the point (1,0) as a starting point and measure arc length around the unit circle in a counterclockwise direction. Imagine a particle which moves with speed one counterclockwise around the circle and is at the point (1,0) at time t=0. It will complete a revolution once every 2 \pi units of time. Thus if the particle is at the point P at time t, it will also be at P at all the times t+2 k \pi, k an integer. Another way to think of the process is to take a copy of the real line, place the origin at the point (1,0), and wrap the line around the circle infinitely many times with the positive direction going counterclockwise. Then each point on the circle will correspond to an infinite family of real numbers spaced 2 \pi apart (Figure 7.1.5).

2024_06_12_6bbb2ed052b26fd5e777g-03.jpg

The Greek letters \theta (theta) and \phi (phi) are often used as variables for angles or circular arc lengths.

Definition

Let P(x, y) be the point at counterclockwise distance 0 around the unit circle starting from (1,0) . x is called the cosine of \theta and y the sine of \theta,

x=\cos \theta, \quad y=\sin \theta \nonumber

2024_06_12_6bbb2ed052b26fd5e777g-04.jpg
Figure 7.1.6

\operatorname{Cos} \theta \cdot and \sin \theta are shown in Figure 7.1.6. Geometrically, if \theta is between 0 and \pi / 2 so that the point P(x, y) is in the first quadrant, then the radius O P is the hypotenuse of a right triangle with a vertical side \sin \theta and horizontal side \cos \theta. By Theorem 1, \sin \theta and \cos \theta are real functions defined on the whole real line. We write \sin ^{n} \theta for (\sin \theta)^{n}, and \cos ^{n} \theta for (\cos \theta)^{n}. By definition (\cos \theta, \sin \theta)=(x, y) is a point on the unit circle x^{2}+y^{2}=1, so we always have

\sin ^{2} \theta+\cos ^{2} \theta=1 \nonumber

Also,

-1 \leq \sin \theta \leq 1, \quad-1 \leq \cos \theta \leq 1 \nonumber

\operatorname{Sin} \theta and \cos \theta are periodic finctions with period 2 \pi. That is,

\begin{aligned} & \sin (\theta+2 \pi n)=\sin \theta \\ & \cos (\theta+2 \pi n)=\cos \theta \end{aligned} \nonumber

for all integers n. The graphs of \sin 0 and \cos \theta are infinitely repeating waves which oscillate between -1 and +1 (Figure 7.1.7).

For infinite values of \theta, the values of \sin \theta and \cos \theta continue to oscillate between -1 and 1 . Thus the limits

\begin{array}{ll} \lim _{\theta \rightarrow \infty} \sin \theta, & \lim _{\theta \rightarrow-\infty} \sin \theta \\ \lim _{\theta \rightarrow \infty} \cos \theta, & \lim _{\theta \rightarrow-\infty} \cos \theta \end{array} \nonumber

do not exist. Figure 7.1.8 shows parts of the hyperreal graph of \sin 0, for positive and negative infinite values of \theta, through infinite telescopes.

The motion of our particle traveling around the unit circle with speed one starting at (1,0) (Figure 7.1.9) has the parametric equations

x=\cos \theta, \quad y=\sin \theta \nonumber

2024_06_12_6bbb2ed052b26fd5e777g-05(1).jpg
Figure 7.1.7
2024_06_12_6bbb2ed052b26fd5e777g-05.jpg
Figure 7.1.8
2024_06_12_6bbb2ed052b26fd5e777g-06.jpg
Figure 7.1.9

The following table shows a few values of \sin \theta and \cos \theta, for \theta in either radians or degrees.

Table 7.1.1
\theta in radians 0 \frac{\pi}{6} \frac{\pi}{4} \frac{\pi}{3} \frac{\pi}{2} \frac{3 \pi}{4} \pi \frac{3 \pi}{2} 2 \pi
\theta in degrees 0^{\circ} 30^{\circ} 45^{\circ} 60^{\circ} 90^{\circ} 135^{\circ} 180^{\circ} 270^{\circ} 360^{\circ}
\sin \theta 0 1 / 2 \sqrt{2} / 2 \sqrt{3} / 2 1 \sqrt{2} / 2 0 -1 0
\cos \theta 1 \sqrt{3} / 2 \sqrt{2} / 2 1 / 2 0 -\sqrt{2} / 2 -1 0 1
Definition

The other trigonometric functions are defined as follows.

\begin{array}{ll} \text { tangent: } & \tan \theta=\frac{\sin \theta}{\cos \theta} \\ \text { cotangent: } & \cot \theta=\frac{\cos \theta}{\sin \theta} \\ \text { secant: } & \sec \theta=\frac{1}{\cos \theta} \\ \text { cosecant: } & \csc \theta=\frac{1}{\sin \theta} \end{array} \nonumber

These functions are defined everywhere except where there is a division by zero. They are periodic with period 2 \pi. Their graphs are shown in Figure 7.1.10.

2024_06_12_6bbb2ed052b26fd5e777g-07(3).jpg
Figure 7.1.10: \tan \theta \cot \theta
2024_06_12_6bbb2ed052b26fd5e777g-07.jpg
Figure 7.1.11

When 0 is strictly between 0 and \pi / 2, trigonometric functions can be described as the ratio of two sides of a right triangle with an angle 0 . Let a be the side opposite \theta, b the side adjacent to \theta, c the hypotenuse as in Figure 7.1.11. Comparing this triangle with a similar triangle whose hypotenuse is a radius of the unit circle, we see that

\begin{array}{lll} \sin \theta=\frac{a}{c}, & \sec \theta=\frac{c}{b}, & \tan \theta=\frac{a}{b} \\ \cos \theta=\frac{b}{c}, & \csc \theta=\frac{c}{a}, & \cot \theta=\frac{b}{a} \end{array} \nonumber

2024_06_12_6bbb2ed052b26fd5e777g-08.jpg
Figure 7.1.12: \csc \theta

Here is a table of trigonometric identities. The diagrams in Figure 7.1.12 suggest possible proofs. ((6) and (7) are called the addition formulas.)

  1. \sin ^{2} \theta+\cos ^{2} \theta=1 \quad (Figure 7.1.12(a))
  2. \tan ^{2} \theta+1=\sec ^{2} \theta \quad (Figure 7.1.12(b))
  3. \cot ^{2} \theta+1=\csc ^{2} \theta \quad (Figure 7.1.12(c))
  4. \sin (-\theta)=-\sin \theta, \quad \cos (-\theta)=\cos \theta (Figure 7.1.12(d))
  5. \sin (\pi / 2-\theta)=\cos \theta, \quad \cos (\pi / 2-\theta)=\sin \theta (Figure 7.1.12(e))
  6. \sin (\theta+\phi)=\sin \theta \cos \phi+\cos \theta \sin \phi (Figure 7.1.12(f))
  7. \cos (\theta+\phi)=\cos \theta \cos \phi-\sin \theta \sin \phi

PROBLEMS FOR SECTION 7.1

In Problems 1-6, derive the given identity using the formula \sin ^{2} \theta+\cos ^{2} \theta=1 and the addition formulas for \sin (\theta+\phi) and \cos (\theta+\phi).

\begin{array}{llll} \mathbf{1} & \tan ^{2} \theta+1=\sec ^{2} \theta & \mathbf{2} & \cos ^{2} \theta+\cos ^{2} \theta \cot ^{2} \theta=\cot ^{2} \theta \\ \mathbf{3} & \sin 2 \theta=2 \sin \theta \cos \theta & \mathbf{4} & \cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta \\ \mathbf{5} & \sin ^{2}\left(\frac{1}{2} \theta\right)=\frac{1-\cos \theta}{2} & 6 & \tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi} \end{array} \nonumber

In Problems 7-10, find all values of \theta for which the given equation is true.

\begin{array}{rlrl} 7 & \sin \theta=\cos \theta & 8 & \sin \theta \cos \theta=0 \\ 9 & \sec \theta=0 & 10 & 5 \sin 3 \theta=0 \end{array} \nonumber

11 Find a value of \theta where \sin 2 \theta is not equal to 2 \sin \theta.

Determine whether the limits exist in Problems 12-17.

12 \lim _{x \rightarrow \infty} \sin x 13 \lim _{x \rightarrow x} \frac{\sin x}{x}
14 \lim _{x \rightarrow \infty} x \sin x 15 \lim _{x \rightarrow 0} x \cos (1 / x)
16 \lim _{x \rightarrow 0} \cot x 17 \lim _{x \rightarrow 0} \tan x

18 Find all values of \theta where \tan \theta is undefined.

19 Find all values of \theta where \csc \theta is undefined.


This page titled 7.1: Trigonometry is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by H. Jerome Keisler.

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