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486 ◆ Chapter 18 18–2 Trigonometric Identities and Equations Sum or Difference of Two Angles Sine of the Sum of Two Angles We wish now to derive a formula for the sine of the sum of two angles, say, and . For example, is it true that sin 20 sin 30 sin 50? Try it on your calculator—you will see that it is not true. We start by drawing two positive acute angles, and (Fig. 18–2), small enough so that their sum ( ) is also acute. From any point P on the terminal side of we draw perpendicular AP to the x axis, and draw perpendicular BP to line OB. Since the angle between two lines equals the angle between the perpendiculars to those two lines, we note that angle APB is equal to . y P + D B A O C x FIGURE 18–2 Then AP AD PD BC PD sin ( ) OP OP OP BC PD OP OP But in triangle OBC, BC OB sin and in triangle PBD, PD PB cos Substituting, we obtain OB sin PB cos sin ( ) OP OP But in triangle OPB, OB OP = cos and PB OP sin Section 18–2 ◆ 487 Sum or Difference of Two Angles Thus: sin ( ) sin cos cos sin The sine of the sum of two angles is not the sum of the sines of each angle. Common Error sin ( ) sin sin Cosine of the Sum of Two Angles Again using Fig. 8–2, we can derive an expression for cos( ). From Eq. 147, OA OC AC OC BD cos ( ) OP OP OP OC BD OP OP Now, in triangle OBC, OC OB cos and in triangle PDB, BD PB sin Substituting, we obtain OB cos PB sin cos ( ) OP OP As before, OB OP cos and PB OP sin Therefore: cos ( ) cos cos sin sin Difference of Two Angles We can obtain a formula for the sine of the difference of two angles merely by substituting for in the equation previously derived for sin( ). sin[ ()] sin cos() cos sin() But for in the first quadrant, () is in the fourth, so cos() cos and sin() sin Therefore, sin ( ) sin cos cos (sin ) which is rewritten as follows: sin ( ) sin cos cos sin We have proven this true for acute angles whose sum is also acute. These identities are, in fact, true for any size angles, positive or negative, although we will not take the space to prove this. 488 Chapter 18 ◆ Trigonometric Identities and Equations We see that the result is identical to the formula for sin( ) except for a change in sign. This enables us to write the two identities in a single expression using the sign. When the double signs are used (such as the symbol), it is understood that the upper signs and the lower signs correspond with the elements in the equation. Sine of Sum or Difference of Two Angles sin( ) sin cos cos sin 167 Similarly, finding cos( ), we have cos[ ()] cos cos() sin sin() Thus: cos ( ) cos cos sin sin or: Cosine of Sum or Difference of Two Angles ◆◆◆ cos( ) cos cos sin sin 168 Example 9: Expand the expression sin (x 3y). Solution: By Eq. 167, sin (x 3y) sin x cos 3y cos x sin 3y ◆◆◆ ◆◆◆ Example 10: Simplify cos 5x cos 3x sin 5x sin 3x Solution: We see that this has a similar form to Eq. 168, with 5x and 3x, so cos 5x cos 3x sin 5x sin 3x cos (5x 3x) cos 8x ◆◆◆ ◆◆◆ Example 11: Prove that cos(180 ) cos Solution: Expanding the left side by means of Eq. 168, we get cos(180 ) cos 180 cos sin 180 sin But cos 180 1 and sin 180 0, so cos(180 ) (1) cos (0) sin cos ◆◆◆ Example 12: Prove that tan x tan y sin (x y) tan x tan y sin (x y) ◆◆◆ Solution: The right side contains functions of the sum of two angles, but the left side contains functions of single angles. We thus start by expanding the right side by using Eq. 167. sin (x y) sin x cos y cos x sin y sin (x y) sin x cos y cos x sin y Our expression now contains sin x, sin y, cos x, and cos y, but we want an expression containing only tan x and tan y. Section 18–2 ◆ 489 Sum or Difference of Two Angles To have tan x instead of sin x, we can divide numerator and denominator by cos x. Similarly, to obtain tan y instead of sin y, we can divide by cos y. We thus divide numerator and denominator by cos x cos y. sin x cos y cos x sin y sin x cos y cos x sin y cos x cos y cos x cos y sin x cos y cos x sin y sin x cos y cos x sin y cos x cos y cos x cos y Then, by Eq. 162, tan x tan y tan x tan y ◆◆◆ Tangent of the Sum or Difference of Two Angles Since, by Eq. 162, sin tan cos we simply divide Eq. 167 by Eq. 168. sin( ) sin cos cos sin tan ( ) cos( ) cos cos sin sin Dividing numerator and denominator by cos cos yields sin cos sin cos tan ( ) sin sin 1 cos cos Applying Eq. 162 again, we get tan tan 1 tan tan A similar derivation (which we will not do) will show that tan( ) is identical to the expression just derived, except, as we might expect, for a reversal of signs. We combine the two expressions using double signs as follows: Tangent of Sum or Difference of Two Angles ◆◆◆ tan cos tan ( ) 1 tan tan 169 Example 13: Simplify tan 3x tan 2x tan 2x tan 3x 1 Solution: This can be put into the form of Eq. 169 if we factor (1) from the denominator. tan 3x tan 2x tan 2x tan 3x 1 tan 3x tan 2x (tan 2x tan 3x 1) tan 3x tan 2x 1 tan 3x tan 2x tan (3x 2x) tan 5x ◆◆◆ 490 Chapter 18 ◆◆◆ ◆ Trigonometric Identities and Equations Example 14: Prove that 1 tan x tan (45 x) 1 tan x Solution: Expanding the left side by Eq. 169, we get tan 45 tan x tan (45 x) 1 tan 45 tan x But tan 45 1, so 1 tan x tan(45 x) 1 tan x ◆◆◆ ◆◆◆ Example 15: Prove that cot y cot x tan (x y) cot x cot y 1 Solution: Using Eq. 152c on the left side yields 1 1 tan y tan x 1 tan x 1 tan y 1 Multiply numerator and denominator by tan x tan y. tan x tan y tan(x y) 1 tan x tan y Then, by Eq. 169: tan (x y) tan (x y) ◆◆◆ Adding a Sine Wave and a Cosine Wave of the Same Frequency In Sec. 17–3, we used vectors to show that the sum of a sine wave and a cosine wave of the same frequency could be written as a single sine wave, at the original frequency, but with some phase angle. The resulting equation is useful in electrical applications. Here we use the formula for the sum or difference of two angles to derive that equation. Let A sin t be a sine wave of amplitude A, and B cos t be a cosine wave of amplitude B, each of frequency /2. If we draw a right triangle (Fig. 18–3) with sides A and B and hypotenuse R, then A R cos R and B R sin B = R sin A = R cos FIGURE 18–3 The sum of the sine wave and the cosine wave is then A sin t B cos t R sin t cos R cos t sin R (sin t cos cos t sin ) the expression on the right will be familiar. It is the relationship of the sine of the sum of two quantities, t and . Continuing, we have A sin t B cos t R (sin t cos cos t sin ) R sin(t ) Section 18–2 ◆ 491 Sum or Difference of Two Angles by Eq. 167. Thus we have the following equation: Addition of a Sine Wave and Cosine Wave ◆◆◆ A sin t B cos t R sin (t ) where B R 兹 A2 B2 and arctan A This is sometimes referred to as magnitude and phase form. 214 Example 16: Express the following as a single sine function: y 3.46 sin t 2.28 cos t Solution: The magnitude of the resultant is R 兹(3.46)2 (2.28)2 4.14 and the phase angle is 2.28 arctan arctan 0.659 33.4 3.46 So y 3.46 sin t 2.28 cos t 4.14 sin(t 33.4) Exercise 2 ◆ Sum or Difference of Two Angles Expand by means of the addition and subtraction formulas, and simplify. 1. sin( 30) 2. cos(45 x) 3. sin(x 60) 4. tan( ) 5. cos p x q 2 6. tan(2x y) 7. sin( 2) 8. cos[( ) ] 9. tan(2 3) Simplify. 10. cos 2x cos 9x sin 2x sin 9x 11. cos( ) sin( ) 12. sin 3 cos 2 cos 3 sin 2 13. sin p x q cos p x q 3 6 Prove each identity. 14. cos x sin(x 90) 15. sin(30 x) cos(60 x) cos x 16. 2 sin cos sin( ) sin( ) 17. sin( 60) cos( 30) 兹 3 cos 18. cos(2 x) cos x ◆◆◆ 492 Chapter 18 ◆ Trigonometric Identities and Equations 19. sin cos(30 ) sin(60 ) 20. cos(x y) cos(x y) 2 cos x cos y 21. cos x sin p x q p sin x q 6 6 22. tan(360 ) tan 23. cos(60 ) sin(330 ) 0 sin 4x cos 4x 24. sin 5x sec x csc x sin x cos x sin(x y) tan x tan y 25. sin(x y) tan x tan y 26. cos(x 60) cos(60 x) cos x cos(x y) 27. tan y cot x sin x cos y 28. cotp x q tanp x q 0 4 4 cos sin 29. tan( 45) cos sin cot cot 1 30. cot( ) cot cot 1 tan x 31. tan p x q 1 tan x 4 Express as a single sine function. 32. y 47.2 sin t 64.9 cos t 33. y 8470 sin t 7360 cos t 34. y 1.83 sin t 2.74 cos t 35. y 84.2 sin t 74.2 cos t 18–3 Functions of Double Angles Sine of 2 An equation for the sine of 2 is easily derived by setting in Eq. 167. sin( ) sin cos cos sin which is rewritten in the following form: Sine of Twice an Angle sin 2 2 sin cos 170 Cosine of 2 Similarly, setting in Eq. 168, we have cos( ) cos cos sin sin which is also given as follows: Cosine of Twice an Angle cos 2 cos2 sin2 171a