Solve 5sinθ+2cosθ=5?

`5sinθ+2cosθ=5`

`=>2cosθ-5(1-sintheta)=0`

`=>2(cos^2(θ/2)-sin^2(theta/2))-5(cos^2(theta/2)+sin^2(theta/2)-2cos(theta/2)sin(theta/2))=0`

`=>2(cos^2(θ/2)-sin^2(theta/2))-5(cos(theta/2)-sin(theta/2))^2=0`

`=>(cos(theta/2)-sin(theta/2))(2cos(theta/2)+2sin(theta/2)-5cos(theta/2)+5sin(theta/2))=0`

`=>(cos(theta/2)-sin(theta/2))(7sin(theta/2)-3cos(theta/2))=0`

When

`(cos(theta/2)-sin(theta/2))=0`

`=>tan(theta/2)=1=tan(pi/4)`

`=>theta/2=npi+pi/4" where "n in ZZ`

`=>theta=2npi+pi/2" where "n in ZZ`

When

`7sin(theta/2)-3cos(theta/2)=0`

`=>tan(theta/2)=3/7=tanalpha` (say)

`=>theta/2=kpi+alpha" where " k in ZZ`

`=>theta=2kpi+2alpha" where " k in ZZ` and `alpha=tan^-1(3/7)`

5sin t + 2cos t = 5
Divide both sides by 5:
`sin t + (2/5)cos t = 1` (1)
Call `tan a = sin a/(cos a) = 2/5` --> `a = 21^@80`, and cos a = 0.93.
The equation (1) becomes;
`sin t cos a + sin a.cos t = cos a`
`sin (t + 21^@80) = 0.93`
Calculator, unit circle and property of sin function give 2 solutions for (t + 21.80):
`(t + 21.80) = 68.43^@`, and
`(t + 21.80) = 180 - 68.43 = 111^@`57#.

a. t + 21.80 = 68.43 +k360
`t = 46^@63 + k360^@`
b. t + 21.80 = 111.57 +k360
`t = 89^@77 + k360^@`, or --> `t = 90^@ + k360^@`
Check by calculator.
t = 46.63 --> 5sin t = 5(0.726) = 3.63 --> 2cos t = 2(0.686) = 1.37
5sin t + 2cos t = 3.63 + 1.37 = 5. Proved
t = 90^@ --> 5sin t = 5(1) = 5 -> cos t = 0.
5sin t + 2cos t = 5 + 0 = 5. Proved

Given that, `5sintheta+2costheta=5`.

`:. 2costheta=5(1-sintheta)`.

`"Squaring, "4cos^2theta=25(1-sintheta)^2`,

`i.e., 4(1-sin^2theta)=25(1-sintheta)^2`

`"Transposing, "4(1-sintheta)(1+sintheta)-25(1-sintheta)^2=0`.

`:. (1-sintheta){4(1+sintheta)-25(1-sintheta)}=0`,

`or, (1-sintheta)(29sintheta-21)=0`.

`:. sintheta=1, or, sintheta=21/29.`

`sintheta=1=sin(pi/2) rArr theta=kpi+(-1)^kpi/2, k in ZZ`.

`sintheta=21/29=sin(arcsin(21/29))," gives, "`

`theta=kpi+(-1)^karcsin(21/29), k in ZZ`.

Altogether, we get ,

`x in {kpi+(-1)^kpi/2}uu{kpi+(-1)^karcsin(21/29)}, k in ZZ`.