Step-by-step explanation:
[tex]4\sin{x} + 3\cos{x} = 4[/tex]
Move the sine term to the right hand side so it becomes
[tex]3\cos{x} = 4 - 4\sin{x}[/tex]
Now take the square of the equation to get
[tex]9\cos^2{x} = 16 - 32\sin{x} + 16\sin^2{x}[/tex]
Use the relation [tex]\cos^2{x} = 1 - \sin^2{x}[/tex] to get
[tex]9 - 9\sin^2{x} = 16 - 32\sin{x} + 16\sin^2{x}[/tex]
Collect all similar terms and we will get
[tex]25\sin^2{x} - 32\sin{x} + 7 = 0[/tex]
Let [tex]y = \sin{x}[/tex] so the above equation becomes
[tex]25y^2 - 32y + 7 = 0[/tex]
Using the quadratic equation, the roots of the above equation are
[tex]y = 1, \frac{7}{25}[/tex]
This means that
[tex] y = \sin{x} = 1 \Rightarrow x = 90°[/tex]
and
[tex]y = \sin{x} = \frac{7}{25} \Rightarrow x = 16.26°[/tex]
