Asked by Annicka
Express each of the following in the form Rsin(x+a), where r>0 and 0< a <2pi
(a) 5cosx + 12sinx
(b) 12cosx + 5sinx
(a) 5cosx + 12sinx
(b) 12cosx + 5sinx
Answers
Answered by
Reiny
I will do the first, and you try the second:
let Rsin(x+a) 5cosx + 12sinx
Rsin(x+a) = R(sinxcosa + cosxsina)
= Rsinxcosa + Rcosxsina
so
Rsinxcosa + Rcosxsina = 5cosx + 12sinx
if this is an identity it must be true for all values of x
Pick any two values of x which you can easily find
let x = 0
Rsin0cosa + Rcos0sina = 5cos0 + 12 sin0
but sin0 = 0 , and cos0 = 1
Rsina = 5
sina = 5/R
let x = 90° or π/2
Rsin90cosa + Rcosasin90 = 5cos90 + 12sin90
sin90 = 1 , and cos90 = 0
Rcosa = 12
cosa = 12/R
but sin^2 a + cos^2 a = 1
25/R^2 + 144/R^2 = 1
R^2 = 169
R = 13
also sina/cosa = (5/R)/(12/R)
tana = 5/12
a = 22.6199°
So <b>5cosx + 12sinx = 13sin(x + 22.6199°)</b>
or if you want radians, set your calculator to radians when you find the tangent inverse
5sinx + 12cosx = 13sin(x + .3948) in radians
check for any angle
e.g. x = 53°
5cos53 + 12sin53 = 12.5927..
13sin(53+22.6199) = 13sin 75.6199 = 12.5927..
not bad
let Rsin(x+a) 5cosx + 12sinx
Rsin(x+a) = R(sinxcosa + cosxsina)
= Rsinxcosa + Rcosxsina
so
Rsinxcosa + Rcosxsina = 5cosx + 12sinx
if this is an identity it must be true for all values of x
Pick any two values of x which you can easily find
let x = 0
Rsin0cosa + Rcos0sina = 5cos0 + 12 sin0
but sin0 = 0 , and cos0 = 1
Rsina = 5
sina = 5/R
let x = 90° or π/2
Rsin90cosa + Rcosasin90 = 5cos90 + 12sin90
sin90 = 1 , and cos90 = 0
Rcosa = 12
cosa = 12/R
but sin^2 a + cos^2 a = 1
25/R^2 + 144/R^2 = 1
R^2 = 169
R = 13
also sina/cosa = (5/R)/(12/R)
tana = 5/12
a = 22.6199°
So <b>5cosx + 12sinx = 13sin(x + 22.6199°)</b>
or if you want radians, set your calculator to radians when you find the tangent inverse
5sinx + 12cosx = 13sin(x + .3948) in radians
check for any angle
e.g. x = 53°
5cos53 + 12sin53 = 12.5927..
13sin(53+22.6199) = 13sin 75.6199 = 12.5927..
not bad
Answered by
Reiny
I noticed a small typo
2nd line clearly should have said
let Rsin(x+a) = 5cosx + 12sinx
2nd line clearly should have said
let Rsin(x+a) = 5cosx + 12sinx
Answered by
[email protected]
12cosx+5sinx=13sin(x+67.38)
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