SCHAUM’S Outline® Trigonometry With Calculator-Based Solutions 5th Edition

CHAPTER 9
Trigonometric Functions of Two Angles

9.1 Addition Formulas

For a proof of these formulas, see Probs. 9.1 to 9.3.

9.2 Subtraction Formulas

For a proof of these formulas, see Prob. 9.4.

9.3 Double-Angle Formulas

For a proof of these formulas, see Prob. 9.14.

9.4 Half-Angle Formulas

For a proof of these formulas, see Prob. 9.15.

SOLVED PROBLEMS

9.1 Prove

and when α and β are positive acute angles.

Let α and β be positive acute angles such that [Fig. 9.1(a)] and [Fig. 9.1(b)].

Fig. 9.1

To construct these figures, place angle α in standard position and then place angle β with its vertex at O and its initial side along the terminal side of angle α. Let P be any point on the terminal side of angle (α + β). Draw PA perpendicular to OX, PB perpendicular to the terminal side of angle α, BC perpendicular to OX, and BD perpendicular to AP.

Now since corresponding sides (OA and AP, and OB and BP) are perpendicular. Then

and

9.2 Show that (1) and (2) of Prob. 9.1 are valid when α and β are any angles.

First check the formulas for the case and . Since

and

the formulas are valid for this case.

Next, it will be shown that if (1) and (2) are valid for any two given angles α and β, the formulas are also valid when, say, α is increased by 90°. Let α and β be two angles for which (1) and (2) hold and consider

and

From the graphs in Sec. 7.3 we see that and . It follows that sin and . Then (a) and (b) reduce to

and

which, by assumption, are valid relations. Thus, (a) and (b) are valid relations.

The same argument may be made to show that if (1) and (2) are valid for two angles α and β, they are also valid when β is increased by 90°. Thus, the formulas are valid when both α and β are increased by 90°. Now any positive angle can be expressed as a multiple of 90° plus θ, where θ is either 0° or an acute angle. Thus, by a finite number of repetitions of the argument, we show that the formulas are valid for any two given positive angles.

It will be left for the reader to carry through the argument when, instead of an increase, there is a decrease of 90° and thus to show that (1) and (2) are valid when one angle is positive and the other negative, and when both are negative.

9.3 Prove