Introduction to Trigonometry Review for Calculus
Here we give a whirlwind review of basic ideas of trigonometry.
When we first learn trigonometry, we do so by studying right triangles and measuring angles in degrees. Look at Fig. 1.25. In calculus, however, it is convenient to study trigonometry in a more general setting, and to measure angles in radians.
Fig. 1.25
Radian Measure of an Angle
Angles will be measured by rotation along the unit circle in the plane, beginning at the positive x -axis. See Fig. 1.26. Counterclockwise rotation corresponds to positive angles, and clockwise rotation corresponds to negative angles. Refer to Fig. 1.27. The radian measure of an angle is defined to be the length of the arc of the unit circle that the angle subtends with the positive x -axis (together with an appropriate + or − sign).
Fig. 1.26
Fig. 1.27
Degree Measure and The Principle of Proportionality
In degree measure, one full rotation about the unit circle is 360°; in radian measure, one full rotation about the circle is just the circumference of the circle or 2π. Let us use the symbol θ to denote an angle. The principle of proportionality now tells us that
In other words
and
Example 1
Sketch the angle with radian measure π/6. Give its equivalent degree measure.
Solution 1
Since
the angle subtends an arc of the unit circle corresponding to 1/12 of the full circumference. Since π/6 > 0, the angle represents a counterclockwise rotation. It is illustrated in Fig. 1.28 .
Fig. 1.28
The degree measure of this angle is
Math Note: In this book we always use radian measure for angles. (The reason is that it makes the formulas of calculus turn out to be simpler.) Thus, for example, if we refer to “the angle 2π/3” then it should be understood that this is an angle in radian measure. See Fig. 1.29.
Fig. 1.29
Likewise, if we refer to the angle 3 it is also understood to be radian measure. We sketch this last angle by noting that 3 is approximately 0.477 of a full rotation 2π—refer to Fig. 1.30.
Fig. 1.30
You Try It: Sketch the angles −2, 1, , 3π/2, 10—all on the same coordinate figure. Of course use radian measure.
Sine and Cosine
Several angles are sketched in Fig. 1.31, and both their radian and degree measures given. If θ is an angle, let ( x , y ) be the coordinates of the terminal point of the corresponding radius (called the terminal radius ) on the unit circle. We call P = ( x , y ) the terminal point corresponding to θ . Look at Fig. 1.32. The number y is called the sine of θ and is written sin θ . The number x is called the cosine of θ and is written cos θ .
Fig. 1.31
Fig. 1.32
Two Fundamental Properties of Sine and Cosine
Since (cos θ , sin θ ) are coordinates of a point on the unit circle, the following two fundamental properties are immediate:
(1) For any number θ ,
(sin θ ) ^{2} + (cos θ ) ^{2} = 1.
(2) For any number θ ,
−1 ≤ cos θ ≤ 1 and −1 ≤ sin θ ≤ 1.
Math Note: It is common to write sin ^{2} θ to mean (sin θ ) ^{2} and cos ^{2} θ to mean (cos θ ) ^{2} .
Example 1
Compute the sine and cosine of π/3.
Solution 1
We sketch the terminal radius and associated triangle (see Fig. 1.33 ). This is a 30–60–90 triangle whose sides have ratios . Thus
Likewise,
It follows that
and
Fig. 1.33
You Try It: The cosine of a certain angle is 2/3. The angle lies in the fourth quadrant. What is the sine of the angle?
Math Note: Notice that if θ is an angle then θ and θ + 2π have the same terminal radius and the same terminal point (for adding 2 just adds one more trip around the circle—look at Fig. 1.34).
As a result,
sin θ = x = sin( θ + 2π)
and
cos θ = y = cos( θ + 2π).
Fig. 1.34
Associated Principle Angle
We say that the sine and cosine functions have period 2π: the functions repeat themselves every 2π units.
In practice, when we calculate the trigonometric functions of an angle θ , we reduce it by multiples of 2π so that we can consider an equivalent angle θ′ , called the associated principal angle, satisfying 0 ≤ θ′ < 2π. For instance,
15π/2 has associated principal angle
3π/2 (since 15π/2 − 3π/2 = 3·2π )
and
− 10π/3 has associated principal angle
2π/3 (since − 10π/3 − 2π/3 = − 12π/3 = −2 ·2π ).
You Try It: What are the principal angles associated with 7π, 11π/2, 8π/3, −14π/5, −16π/7?
Sine, Cosine, and Right Triangle Trigonometry
What does the concept of angle and sine and cosine that we have presented here have to do with the classical notion using triangles? Notice that any angle θ such that 0 ≤ θ < π/2 has associated to it a right triangle in the first quadrant, with vertex on the unit circle, such that the base is the segment connecting (0, 0) to ( x , 0) and the height is the segment connecting ( x , 0) to ( x , y ). See Fig. 1.35.
Fig. 1.35
Then
and
Thus, for angles θ between 0 and π/2, the new definition of sine and cosine using the unit circle is clearly equivalent to the classical definition using adjacent and opposite sides and the hypotenuse. For other angles θ , the classical approach is to reduce to this special case by subtracting multiples of π/2. Our approach using the unit circle is considerably clearer because it makes the signatures of sine and cosine obvious.
Tangent, Cotangent, Secant, and Cosecant
Besides sine and cosine, there are four other trigonometric functions:
Whereas sine and cosine have domain the entire real line, we notice that tan θ and sec θ are undefined at odd multiples of π/2 (because cosine will vanish there) and cot θ and csc θ are undefined at even multiples of π/2 (because sine will vanish there). The graphs of the six trigonometric functions are shown in Fig. 1.36.
Fig. 1.36(a) Graphs of y = sin x and y = cos x .
Fig. 1.36(b) Graphs of y = tan x and y = cot x .
Fig. 1.36(c) Graphs of y = sec x and y = csc x .
Example 1
Compute all the trigonometric functions for the angle θ = 11π/4.
Solution 1
We first notice that the principal associated angle is 3π/4, so we deal with that angle. Figure 1.37 shows that the triangle associated to this angle is an isosceles right triangle with hypotenuse 1.
Fig. 1.37
Therefore and . It follows that
Similar calculations allow us to complete the following table for the values of the trigonometric functions at the principal angles which are multiples of π/6 or π/4.
Fundamental Properties for Trigonometry Functions
(1) For any number θ ,
(sin θ ) ^{2} + (cos θ ) ^{2} = 1.
(2) For any number θ ,
−1 ≤ cos θ ≤ 1 and −1 ≤ sin θ ≤ 1.
Besides properties (1) and (2) above, there are certain identities which are fundamental to our study of the trigonometric functions. Here are the principal ones:
Example 1
Prove identity number (3).
Solution 1
We have
You Try It: Use identities (11) and (12) to calculate cos(π/12) and sin(π/12).
Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.
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From Calculus Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.