Higher Mathematics UNIT OUTCOME 4 The Wave Equation Contents The Wave Equation 146 1 Expressing pcosx + qsinx in the form kcos(x a 146 Expressing pcosx + qsinx in other forms 147 Multiple Angles 148 4 Maximum and Minimum Values 149 Solving Equations 10 6 Sketching Graphs of y pcosx + qsinx 1 This document was produced specially for the HSNuknet website, and we require that any copies or derivative works attribute the work to Higher Still Notes For more details about the copyright on these notes, please see http://creativecommonsorg/licenses/by-nc-sa//scotland/
OUTCOME 4 The Wave Equation 1 Expressing pcosx + qsinx in the form kcos(x a An expression of the form p cos x + q sin x can be written in the form k cos( x a where: k sin a k p + q and tana The following example shows how to achieve this EXAMPLES 1 Write cos x + 1 sin x in the form k cos( x where 0 a 60 Step 1 Expand k cos( x a using the cos x + 1sin x compound angle formula k cos( x Step Rearrange to compare with p cos x + q sin x Step Compare the coefficients of cos x and sin x with p cos x + qsin x Step 4 Mark the quadrants on a CAST diagram, according to the signs of k cos a and k sin a Step Find k and a using the formulae above (a lies in the quadrant marked twice in Step 4 Step 6 State p cos x + q sin x in the form k cos( x a using these values k cos x cos + k sin x sin k cos cos x + k sin sin x 1 k cos k sin 1 180 + 60 k + 1 1 169 k sin tan k cos 1 1 a tan 1 67 4 (to 1 d p cos x + 1sin x 1cos x 67 4 Page 146
Write cos x sin x in the form k cos( x a where 0 a π cos x sin x k cos( x a k sin a π a a S A π + a π a fourth quadrant k cos x cosa + k sin x sina cos x + k sin a sin x k + ( 4 Hence cos sin 4 cos( 74 x x x Expressing pcosx + qsinx in other forms k sin a tana k cos a First quadrant answer is: 1 tan 0 40 (to d p So a π 0 40 74 (to d p An expression in the form p cos x + q sin x can also be written in any of the following forms using a similar method EXAMPLES k cos( x + a k sin( x a k sin( x + a 1 Write 4cos x + sin x in the form k sin( x + where 0 a 60 4 cos x + sin x k sin( x + k cos k sin 4 180 + 60 k sin x cos + k cos x sin k cos sin x + k sin cos x k 4 + Hence 4cos x + sin x sin( x + 1 k sin tan k cos 4 1 a tan 4 1 (to 1 d p Page 147
Write cos x sin x in the form k cos( x + a where 0 a π cos x sin x k cos( x + a 1 k sin a π a a S A π + a π a k cos x cos a k sin x sin a k cos a cos x k sin a sin x k 1 + 1+ Hence cos x sin x cos( x π Multiple Angles + 4 k sin a tana k cos a a tan π 1 The same method is used with expressions of the form pcos( nx + qsin( nx, where n is a constant EXAMPLE Write cosx + 1 sin x in the form k sin( x + where 0 a 60 cosx + 1sinx k sin( x + k cos 1 k sin 180 + 60 k sin x cos + k cos x sin k cos sin x + k sin cos x k 1 + 1 169 Hence cos x + 1sin x 1sin( x + 6 k sin tan k cos 1 1 a tan 1 6 (to 1 d p Page 148
4 Maximum and Minimum Values To work out the maximum or minimum values of p cos x + qsin x, we can rewrite it as a single trigonometric function, eg k cos( x a Recall that the maximum value of the sine and cosine functions is 1, and their minimum is 1 y y sin x y y cos x 1 max 1 1 max 1 O 1 EXAMPLE Write 4 sin x + cos x in the form k cos( x a where 0 a π and state: (i the maximum value and the value of 0 x < π at which it occurs (ii the minimum value and the value of 0 x < π at which it occurs 4sin x + cos x k cos( x a 1 k sin a 4 π a π a S A π + a π a k cos x cosa + k sin x sin a cos x + k sin a sin x k ( 1 + 4 17 O 1 Hence 4 sin x + cos x 17 cos( x 1 6 x min 1 π x min 1 k sin a tana 4 1 a tan ( 4 1 6 (to d p The maximum value occurs when: cos( x 1 6 1 1 x 1 6 cos ( 1 x 1 6 0 x 1 6 (to d p The minimum value occurs when: cos( x 1 6 1 1 x 1 6 cos ( 1 x 1 6 π x 4 468 (to d p Page 149
Solving Equations The wave equation can be used to help solve trigonometric equations involving both a sin( nx and a cos( nx term EXAMPLES 1 Solve cos x + sin x where 0 x 60 First, we write cos x + sin x in the form k cos( x : cos x + sin x k cos( x k cos k sin 1 180 + 60 k cos x cos + k sin x sin k cos cos x + k sin sin x k + 1 6 Hence cos x + sin x 6 cos( x 11 Now we use this to help solve the equation: cos x + sin x 6 cos( x 11 cos( x 11 6 x 11 66 9 or 60 66 9 x 11 66 9 or 9 1 x 78 or 04 4 180 x k sin tan 1 k cos 180 + x 60 x 1 a tan 1 11 (to 1 d p x S A 1 x 11 cos 6 66 9 (to d p Page 10
Solve cosx + sinx 1 where 0 x π First, we write cos x + sin x in the form k cos( x a : cosx + sin x k cos( x a k sin a π a a S A π + a π a k cos x cos a + k sinx sin a k cos a cos x + k sina sinx k + ( 4 + 9 1 Hence cosx + sinx 1 cos( x 0 98 Now we use this to help solve the equation: k sin a tana k cos a 1 a tan 0 98 (to d p cosx + sin x 1 π x x S A 0 < x < π 1 cos( x 0 98 1 cos( x 0 98 1 1 π + x 0 < x < 4π π x 1 x 0 98 cos 1 ( 1 1 90 (to d p x 0 98 1 90 or π 1 90 or π + 1 90 or π + π 1 90 or π + π + 1 90 x 0 98 1 90 or 4 99 or 7 7 or 11 76 x 7 or 976 or 8 6 or 1 9 x 1 17 or 988 or 4 78 or 6 10 Page 11
6 Sketching Graphs of y pcosx + qsinx Expressing p cos x + q sin x in the form k cos( x a enables us to sketch the graph of y pcos x + q sin x EXAMPLES 1 (a Write 7cos x + 6sin x in the form k cos( x, 0 a 60 (b Hence sketch the graph of y 7cos x + 6sin x for 0 x 60 (a First, we write 7 cos x + 6sin x in the form k cos( x : 7 cos x + 6sin x k cos( x k cos 7 k sin 6 180 + 60 k cos x cos + k sin x sin k cos cos x + k sin sin x k 6 + 7 6 + 49 8 x x x Hence 7 cos + 6sin 8 cos( 40 6 k sin tan k cos (b Now we can sketch the graph of y 7cos x + 6sin x : y y 7 cos x + 6sin x 84 6 7 1 a tan 6 7 40 6 (to 1 d p O x 84 40 60 60 Page 1
Sketch the graph of y sin x + cos x for 0 x 60 First, we write sin x + cos x in the form k cos( x : sin x + cos x k cos( x k cos k sin 1 180 + 60 k cos x cos + k sin x sin k cos cos x + k sin sin x k + 1 1+ Hence sin x + cos x cos( x 0 k sin tan k cos a tan 0 1 1 1 Now we can sketch the graph of y sin x + cos x : y y sin x + cos x O x 0 60 Page 1
(a Write sin x 11cos x in the form k sin( x a, 0 a 60 (b Hence sketch the graph of y sin x 11cos x +, 0 x 60 (a sin x 11cos x k sin( x k cos k sin 11 180 + 60 k sin x cos + k cos x sin k cos sin x + k sin cos x k + 11 + 11 6 6 Hence sin x 11cos x 6sin( x 6 (b Now sketch the graph of y sin x 11cos x + 6sin( x 6 + : y 8 6 y sin x 11 cos x + k sin tan k cos ( 1 a tan 11 11 6 (to 1 d p O 4 6 6 60 x Page 14