Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 6: The Trigonometric Functions And Their Inverses – Section 6.1.1: The Trigonometric Functions

 

6.1.1
The Trigonometric Functions

 

 

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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. 1.2

 

Trigonometric ratios of an acute angle of a right triangle are defined as
various ratios of the sides and hypotenuse of triangle.

 

The word  trigonometry  comes from the Greek words: tri, which means three, gono, which means angle, and metria,
which means measurement.

 

Arbitrary Triangles

 

 

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 x-axis 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. 2.1

 

A 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 non-negative real number. Then the measure of this angle is r o. That's why the
angle intercepting any entire circle measures 360o , that intercepting any half-circle measures 180o, and a right angle
measures 90o.

 

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. 2.3

 

A radian is equal to the radius.

 

 

Conversion From Degree To Radian And Vice Versa

 

 

Example 2.1

 

 

Solution

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. 3.1

 

The 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. 4.1

 

Any 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,
cos x = u.

 

 

Note that sine is on the vertical v-axis and cosine on the horizontal u-axis.

 

 

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 v-coordinate 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. 6.3

 

Extensions 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.

 

Example 6.1

 

Show that:

 

 

Solution

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 z-axis and cotangent is on the horizontal w-axis.

 

 

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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 u-axis (same as cosine) and cosecant (= 1/sine) is on the vertical
v-axis (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

 

 

 

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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

 

 

 

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Solution

 

 

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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:

 

 

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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:

 

 

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5.  Show that:

 

   

 

Solution

 

 

 

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