Download presentation

1
**Expressing asinx + bcosx in the forms**

Rsin(x ± a) or Rcos(x ± a) The graph below is y = 3cosx + 4sinx. This can be considered as either a sine or a cosine graph which has been translated horizontally and stretched vertically.

2
**If it is considered to be a cosine curve then it has**

been translated horizontally and stretched vertically by a factor of 530 5 The object is to find the horizontal translation and vertical stretch.

3
**If it is considered to be a sine curve then it has**

been translated horizontally and stretched vertically by a factor of -370 5 The object is to find the horizontal translation and vertical stretch.

4
**If the curve is taken to be a translated cosine curve then its equation will be of the form**

3cosx + 4sinx = Rcos(x - a) Where a is the horizontal translation And R is the vertical stretch Note the question contains a PLUS in the middle and the translated equation contains a MINUS because the curve is translated in a positive x direction.

5
3 cosx + sinx = Rcos(x - a) = R(cosxcosa + sinxsina) Using cos(A - B) cosx Matching up the left and right hand side then Rcosa = 3 Rsina = 4 4 = (Rcosa) + (Rsina) sinx

6
Rsina = 4 Rcosa = 3 Dividing these two equations a = 53.1o

7
Rsina = 4 Rcosa = 3 Squaring these two equations R2 sin2a = and R2 cos2a = 9 Adding these two equations R2 sin2a + R2cos2a = = 25 R2(sin2a + cos2a) = 25 R = 5 as sin2a + cos2a = 1

8
**Hence 3cosx + 4sinx = Rcos(x - a)**

This is a cosine graph which has been translated horizontally and stretched vertically by a factor of 5 This is evident from the graph on the right

9
**If the curve is taken to be a translated sine curve then its equation will be of the form**

3cosx + 4sinx = Rsin(x + a) Where a is the horizontal translation And R is the vertical stretch Note the question contains a PLUS in the middle and the translated equation contains a PLUS because the curve is translated in a negative x direction.

10
3 cosx + sinx = Rsin(x + a) = R(sinxcosa + cosxsina) Using sin(A + B) sinx Matching up the left and right hand side then Rcosa = 4 Rsina = 3 4 = (Rcosa) + (Rsina) cosx

11
Rsina = 3 Rcosa = 4 Dividing these two equations a = 36.9o

12
Rsina = 3 Rcosa = 4 Squaring these two equations R2 sin2a = 9 and R2 cos2a = 16 Adding these two equations R2 sin2a + R2cos2a = = 25 R2(sin2a + cos2a) = 25 R = 5 as sin2a + cos2a = 1

13
**Hence 3cosx + 4sinx = Rsin(x + a)**

This is a sine graph which has been translated horizontally –36.90 and stretched vertically by a factor of 5 This is evident from the graph on the right

14
**Using the Rsin(x ± a) or Rcos(x ± a) form **

to solve equations of the form acosx + bsinx = c Solve cosx + 4sinx = 4 3cosx + 4sinx = 5cos(x ) Shown previously So 5cos(x – 53.1) = 4 Let y = x – 53.1 So cosy = y = cos-1 = 36.8, -36.8, 323.1 x – 53.1 = 36.8, -36.8, 323.1 find 1st two answers and add 360 x = 89.9, 16.3, add 53.1 to both sides

Similar presentations

OK

EXAMPLE 1 Solve a simple absolute value equation Solve |x – 5| = 7. Graph the solution. SOLUTION | x – 5 | = 7 x – 5 = – 7 or x – 5 = 7 x = 5 – 7 or x.

EXAMPLE 1 Solve a simple absolute value equation Solve |x – 5| = 7. Graph the solution. SOLUTION | x – 5 | = 7 x – 5 = – 7 or x – 5 = 7 x = 5 – 7 or x.

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google