Calculus Of One Real Variable – By Pheng Kim Ving

6.1.1 
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1. Trigonometric Ratios 
Fig. 1.1 
word “ sign”. Similarly, the ratios:
The three ratios sine, cosine, and
tangent are the primary trigonometric ratios. There are also three other
ratios.
They are cotangent, secant, and cosecant, denoted by cot,
sec, and csc respectively. They're defined as the
reciprocals of
tangent, cosine, and sine respectively, and are thus called, well, the reciprocal
trigonometric ratios.
Fig. 1.2Trigonometric ratios of an acute angle of a right triangle are defined as

The word trigonometry comes from the Greek
words: tri, which means three, gono, which means angle, and metria,
which means measurement.
Fig. 1.3 Trigonometric Ratios In An Arbitrary Triangle. 
Fig. 1.4 Trigonometric Ratios In An Arbitrary Triangle. 
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2. The Radian Measure 
The Degree
Consider a circle of radius r centered at the origin 0 of the xaxis and intersecting its positive side at A. See Fig. 2.1.
Let's divide the circumference of the circle into 360 equal parts or arcs,
starting from A. Consider an angle whose
vertex
Fig. 2.1A degree is equal to one 360th of the circumference of a circle. 
is at the centre of the circle
and which intercepts one of the arcs. Clearly, the length of the arc depends on
the value of r,
but the size of the angle doesn't. The size of this angle constitutes a unit of
the measurement of angles and is called the
degree. So the degree, denoted by deg or ^{o}, is defined to
be equal to one 360th of the circumference of a
circle.
Suppose an angle intercepts an
arc of a circle centered at its vertex such that the
length of the arc is r 360ths of
the
circumference of the circle, where r is a nonnegative
real number. Then the measure of this angle is r ^{o}. That's why
the
angle intercepting any entire circle measures 360^{o} , that intercepting any
halfcircle measures 180^{o}, and a right angle
measures 90^{o}.
The Grad
Similarly, if we divide the circumference
of a circle into 400 equal arcs, then we get another unit of the measurement of
angles, called the grad. So the grad is defined to be one 400th of the
circumference of a circle.
The Radian
Fig. 2.2


Fig. 2.3A radian is equal to the radius.

Conversion From Degree To Radian And Vice Versa
EOS
Omission Of The Unit Radians In Writing
Remark 2.1
Units of measurements of angles
are defined by using, not the normal units of length like the meter, but elements
of the
circle like one 360th of the circumference for the degree or the radius for the
radian. Also see Problem & Solution 1.
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3. The Unit Circle 
Fig. 3.1The unit circle (red) and the circle of radius r. 
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4. Angles And The Unit Circle 
For the unit circle, the radian measure of a central angle is equal to the length of the intercepted arc. Any real number can be regarded as the radian measure of a signed angle. 
Fig. 4.1Any real number x can be
regarded as the radian measure of a signed angle.

The line OA
is called the initial arm and the line OP
the terminal arm of the angle x. We say
that x terminates in the
quadrant where the terminal arm is; eg, if the terminal arm is in the 3rd
quadrant, then we say that x terminates
in the
3rd quadrant.
Sizes Of Angles
The angles:

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5. Trigonometric Functions 
In Part 1 we defined the sine,
cosine, tangent, cotangent, secant, and cosecant of acute angles, using the
right triangle,
and they're called the trigonometric ratios. Now we're going to extend them to
all angles, employing the unit circle, and
they're called the trigonometric functions. For the rest of this
section, every circle is the unit circle centered at the origin of
the uv coordinate system, A is the point (1, 0), and B
is the point (0, 1), unless stated otherwise.
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6. The Sine And Cosine Functions 
Let x be an acute
angle, OP its terminal arm, and (u, v) the
coordinates of P. See Fig. 6.1. We have sin
x = UP/OP =
v/1 = v and cos x = OU/OP = u/1 = u. We extend the sine and
cosine to all angles as follows. Let x be any angle,
OP its terminal arm, and (u,
v) the coordinates of P.
Then the sine and cosine of x are defined
as follows:
sin x = v, 
Note that sine is on the vertical vaxis and cosine on the horizontal uaxis.
Each real number x is mapped to a unique value v = sin x. Hence,
this mapping is a function. It's called the sine
function. Its domain is R.
Its range is [–1, 1], because the vcoordinate of P always falls in [–1, 1] no matter
what value
Fig. 6.1 
x has. Similarly, u = cos x defines the cosine function, whose domain is R and range is [–1, 1] also.
Justification Of The Extensions
Fig. 6.2 
Fig. 6.3Extensions of sine and cosine to all angles using unit circle are justified. 
We've just justified the extension of sine to all angles,
positive or 0 or negative. The justification of the extension of
cosine is similar.
Show that:
Fig. 6.4

EOS
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7. The Tangent And Cotangent Functions 
Let x be any
angle. The extended tangent function and cotangent function are
defined by tan x = (sin
x)/(cos x)
and cot x = 1/(tan
x) = (cos x)/(sin
x) respectively.
is left as Problem & Solution 3. The cases for the remaining two quadrants are similar to it. We have:

Fig. 7.1 
Note that tangent is on the vertical zaxis and cotangent is on the horizontal waxis.
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8. The Secant And Cosecant Functions 
Let x
be any angle. The extended secant function and cosecant function
are defined by sec x = 1/(cos x) and
csc x = 1/(sin x)
respectively.
is left as Problem & Solution 4. The cases
for the remaining two quadrants are similar to it. We have:

Fig. 8.1 
Note that secant (= 1/cosine) is on the horizontal uaxis
(same as cosine) and cosecant (= 1/sine) is on the vertical
vaxis (same as sine).
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9. Trigonometric Or Circular Functions 
The trigonometric functions sine, cosine, tangent, cotangent,
secant, and cosecant are defined by means of a circle.
For this reason, they're also called circular functions. They're defined
using the unit circle, as summarized in Fig. 9.1.
Fig. 9.1 
Problems & Solutions 
1. Explain
why lengths of arcs of circles in a normal unit of length like the meter cannot
be used as a measurement of
angles, unless the unit circle (circle
of radius 1) is adopted for use. Also see Remark 2.1.
Solution
Solution
3. Suppose the angle x
terminates in the 2nd quadrant. See the figure below. Show that tan
x = z and cot x = w. Also
see the discussion following Eqs. [7.1] and [7.2].
Solution
Right triangles OPU and OZA are similar. We have:
4. Suppose the angle x
terminates in the 2nd quadrant. See the figure below. Show that sec
x = q and csc x = r. Also
see the discussion following Eqs. [8.1] and [8.2].
Solution
Right triangles OPU and OQP are similar. We have:
5. Show that:
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