7.1 TRIGONOMETRY
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 form 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.
Figure7.1.1
Trigonometryis based on the notion of the length of an arc. Lengths of curveswere introduced in Section 6.3. Although
that section provides auseful background, this chapter can also be studied independently ofChapter6. As a staring point we shall give a formula for the lengthof an arc in terms of the area of a sector. (This formula wasproved as a theorem in Section 6.3 but
can also be taken as thedefinition of arc length.)
DEFINITON
Thelength of an arc PQ on the unit circle is equal to twice thearea of the sector POQ, s=2A.
Thisformula can be seen intuitively as follows. Consider a small arc PQof length △s (Figure 7.1.2). The sector POQ
is a thin wedge whichis almost a right triangle of altitude one and base △s. Thus △A∼12△ s. Making △s infinitesimal and adding up, we get A=12s.
Thenumber π ~ 3.14159 is defined as the area of the unit circle. Thusthe
Unitcircle has circumference 2π.
Thearea of a sector POQ is a defined integral. For example, if P is thepoint p(1.0) and the point Q(x,y) is in the
first quadrant, then wesee form Figure 7.1.3 that the are.
Ax=12x1-x2+x11-t2dt.
Noticethat A(x) is a continuous function of x.The length of an arc has thefollowing basic property.
Figure7.1.2 Figure 7.1.3
THEOREM 1
Let Pbe the point P(1,0). For every number s between 0 and 2π
thereis a point
Qon the unit circle such that the are PQ has length s.
PROOFWe give the proof for s between 0 and π2, whence
0 ≤12 s ≤π4.
Let A(x)be the area of the sector POQ where Q = Q(x,y) (Figure 7.1.4). ThenA(0) = π4, A(1)=0 and the function A(x) is continuous for 0≤ x≤1. By the Intermediate
Value Theorem there is a point x0 between 0 and 1where the sector has area 12s,
A(x0)=12s.
Thereforethe arc PQ has length
2A(x0)= S.
Figure7.1.4
Arc lengthsare used to measure angles. Two units of measurement for angles
Areradians (best for mathematics) and degrees (used in everyday life).
DEFINITION
Let Pand Q be two points on the unit circle. The measure of the angle ∠POQ
in radiansis the length of the arc PQ. A degree is defined as
10= π180 radians,
whencethe measure of ∠POQ in degrees is 180π times the length of PQ.
Approximately, 10 ~ 0.01745 radians,
1 radian~ 57018' =(571860)0.
Acomplete revolution is 3600 or 2π radians. A straight angle is 1800or π radians.A
Rightangle is 900 or π2 radians.
It isconvenient to take the point (1,0) as a starting point and measurearc length around the unit circle in a counterclockwise
direction.Imagine a particle which moves wish speed one counterclockwise aroundthe circle and is at the point (1,0)at time t=0. It will complete arevolution once every 2π units of time. Thus if the particle is atthe point P at time t, it will also be at P
at all the times t+2kπ,k an integer. Another way to think of the process is to take a copyof the real line, place the origin at the point(1,0), and wrap theline around the circle infinitely many times with the positivedirection going counterclockwise. Then
each point on the circle willcorrespond to an infinite family of real numbers spaced 2πapart(Figure 7.1.5).
TheGreek letters θ (theta) and ∅ (phi) are often used as variablesfor angles or circular arc lengths.
DEFINITION
LetP(x, y) be the point at counterclockwise distance θ around the unitcircle staring from (1, 0). x is called the cosine of θ and y
thesine of θ,
x=cosθ, y = sinθ.
Cosθ· and sin θ are shown in Figure 7.1.6. Geometrically, if θ isbetween 0 and π2 so that the point P(x,y) is in the first quadrant,then
the radius OP is the hypotenuse of a right triangle with avertical side sin θ and horizontal side cos θ. By Theorem 1, sin θand cos θ are real functions defined on the whole real line. Wewrite sinn0 for sin θn, and cosn θ for cos θn. By definition(sinθ,cosθ)
=(x,y) is a point on the unit circle x2+y2=1, so wealways have
sin2θ+cos2θ=1.
Also, -1 ≤ sin θ ≤1, -1 ≤ cos θ ≤1.
Sinθ and cos θ are periodic functions with period 2π. That is,
sin(θ+2πn)= sinθ,
cos(θ+2πn)= cosθ
forall integers n. The graphs of sin 0 and cos 0 are infinitelyrepeating waves which oscillate between -1 and +1 (Figure 7.1.7).
Forinfinite values of θ, the values of sin 0 and cos 0 continue tooscillate between -1 and 1. Thus the limits
donot exist. Figure 7.1.8 shows parts of the hyperreal graph of sin 0,for positive and negative infinite values of 0, through infinitetelescopes.
Themotion of our particle traveling around the unit circle with speedone
Startingat (1,0)(Figure 7.1.9)has the parametric equations
x=cosθ, y = sinθ.
Thefollowing table shows a few values of sin θ and cosθ, for θ ineither radians or degrees.
DEFINITION
Theother trigonometric functions are defined as follows.
tangent: tan θ=sinθcosθ
cotangent: cosθ=cosθsinθ
secant: secθ=1cosθ
cosecant: cscθ=1sinθ
Thesefunctions are defined everywhere except where there is a division byzero. They are periodic with period 2π. Their graphs are shown
inFigure 7.1.10.
When0 is strictly between 0 and π/2, trigonometric functions canbe described as the ratio of two sides of a right triangle
with anangle 0. Let a be the side opposite 0, b the side adjacent to θ, cthe hypotenuse as in Figure 7.1.11. Comparing this triangle
witha
similar triangle whose hypotenuse is a radius of the unit circle,we see that
sinθ=ac, secθ=cb, tanθ=ab,
cosθ=bc, cscθ=ca, cotθ=ba.
Hereis a table of trigonometric identities. The diagrams in Figure 7.1.12suggest possible.( ⑹ and ⑺ are called the addition formulas.)
⑴sin2θ+cos2 θ=1 (Figure 7.1.12(a))
⑵ tan2θ+1 = sec2 θ (Figure 7.1.12(b))
⑶ cot2θ+1 = csc2 θ (Figure 7.1.12(c))
⑷sin-θ=-sinθ,cos-θ=-cosθ (Figure 7.1.12(d))
⑸sinπ/2-θ=cosθ,cosπ/2-θ)=sinθ (Figure 7.1.12(e))
⑹sinθ+∅=sinθcos∅+cosθsin∅ (Figure 7.1.12(f))
⑺cosθ+∅=cosθcos∅-sinθsin∅ (Figure 7.1.12(f))
PROBLEMSFOR SECTION 7.1
InProblems 1-6, derive the given identity using the formula sin2 θ+cos2θ=1 and the addition formulas for sinθ+∅and
cos θ+∅.
1 tan2 θ+1 = sec2 θ 2 cos2 θ+cos2 θcot2 θ=cot2 θ
3 sin2θ=2sinθcosθ 4 cos2θ=cos2 θ-sin2 θ
5 sin2(12θ)= 1-cosθ2 6 tan θ+∅= tanθ+cos∅ 1-tanθ+cos∅
InProblems 7-10, find all values of θ for which the given equation istrue.
7 sinθ=cosθ 8 sinθcosθ=0
9 secθ=0 10 5 sin 3θ=0
11 Find a value of θ where sin 2θ is not equal to 2 sin θ.
Determinewhether the limits exist in Problems 12-17.
18 Find all values of θ where tan θ is undefined.
19 Find all values of θ where csc θ is undefined.