We have: 2 sin^(2)(x) + 3 sin(x) = 42sin2(x)+3sin(x)=4; [0, 360^(circ))[0,360∘)
If you take a look at the equation, you may notice that it is almost in the form of a quadratic equation.
Let's subtract 44 from both sides of the equation:
Rightarrow 2 sin^(2)(x) + 3 sin(x) - 4 = 0⇒2sin2(x)+3sin(x)−4=0
We can consider sin(x)sin(x) to be a variable of its own, so let's apply the quadratic formula:
Rightarrow sin(x) = frac(- 3 pm sqrt(3^(2) - 4(2)(- 4)))(2(2))⇒sin(x)=−3±√32−4(2)(−4)2(2)
Rightarrow sin(x) = frac(- 3 pm sqrt(9 + 32))(4)⇒sin(x)=−3±√9+324
Rightarrow sin(x) = frac(- 3 pm sqrt(41))(4)⇒sin(x)=−3±√414
Rightarrow sin(x) = frac(- 3 pm 6.40)(4)⇒sin(x)=−3±6.404
Rightarrow sin(x) = - 2.35, 0.85⇒sin(x)=−2.35,0.85
However, the range of sin(x)sin(x) is [- 1, 1][−1,1].
So sin(x) = 0.85sin(x)=0.85.
Let's set the reference angle as sin(x) = 0.85 Rightarrow x = 58^(circ)sin(x)=0.85⇒x=58∘:
Rightarrow sin(x) = 0.85⇒sin(x)=0.85
Rightarrow x = 58^(circ), 180^(circ) - 58^(circ)⇒x=58∘,180∘−58∘
therefore x = 58^(circ), 122^(circ)
Therefore, the solutions to the equation are x = 58^(circ) and x = 122^(circ).
