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Trigonometry Review for Calculus Help

By — McGraw-Hill Professional
Updated on Sep 1, 2011

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.

Basics 1.7 Trigonometry

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).

Basics 1.7 Trigonometry

Fig. 1.26

Basics 1.7 Trigonometry

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

Basics 1.7 Trigonometry

In other words

Basics 1.7 Trigonometry

and

Basics 1.7 Trigonometry

Example 1

Sketch the angle with radian measure π/6. Give its equivalent degree measure.

Solution 1

Since

Basics 1.7 Trigonometry

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 .

Basics 1.7 Trigonometry

Fig. 1.28

The degree measure of this angle is

Basics 1.7 Trigonometry

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.

Basics 1.7 Trigonometry

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.

 

Basics 1.7 Trigonometry

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 .

Basics 1.7 Trigonometry

Fig. 1.31

Basics 1.7 Trigonometry

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 .

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