Im having problems solving this equation:
2sin(x) = cos(x) + 2
I keep getting an imaginary number..:/
2 sin x -2 = cos x = sqrt(1 - sin^2 x)
4 sin^2 x - 8 sin x + 4 = 1 - sin^2 x
5 sin^2 x - 8 sin x + 3 = 0
Let sin x = y
5y^2 -8y +3 = 0
(5y-3)(y-1) = 0
sin x = 1 or 3/5
x = pi/2 (90 degrees) is one solution. There are others as well
im a bit confused as to where the 8sin(x) came from
It somes from squaring 2sin x - 2.
[2(sin x -1)]^2 = 4(sinx-1)^2
= 4 (sin^2 - 2x +1)= ...
ah, thank you, i don't know why i was squaring it differently (wrong way)
Jess, don't forget that after you square an equation all answers you obtain must be verified.
for example, the second part to drwls solution was
sinx = 3/5
which produces answers of 36.87º and 143.13º
when these are checked in the original equation, x= 143.13 works but x = 36.87 does not
so x = 90º or x = 143.13º
To solve the equation 2sin(x) = cos(x) + 2, we need to use trigonometric identities and techniques. Let's break down the steps:
Step 1: Rearrange the equation.
Start by subtracting cos(x) from both sides to isolate sin(x):
2sin(x) - cos(x) = 2.
Step 2: Apply the Pythagorean identity.
The Pythagorean identity states that sin^2(x) + cos^2(x) = 1. Rearranging this equation, we get 1 - sin^2(x) = cos^2(x).
Step 3: Substitute the Pythagorean identity into the rearranged equation.
Replace cos^2(x) in the rearranged equation (2sin(x) - cos(x) = 2) with 1 - sin^2(x):
2sin(x) - √(1 - sin^2(x)) = 2.
Step 4: Solve for sin(x).
To eliminate the square root, we can square both sides of the equation:
(2sin(x) - √(1 - sin^2(x)))^2 = 2^2.
Simplifying the equation:
4sin^2(x) - 4sin(x)√(1 - sin^2(x)) + (1 - sin^2(x)) = 4.
Expanding and collecting like terms:
5sin^2(x) - 4sin(x)√(1 - sin^2(x)) + 1 = 4.
Step 5: Rearrange and rewrite the equation.
Move all terms to one side of the equation:
5sin^2(x) - 4sin(x)√(1 - sin^2(x)) - 3 = 0.
Step 6: Identify a substitution.
Let's make a substitution: Let u = sin(x). This allows us to write the equation as a quadratic equation in terms of u.
Replace sin^2(x) with u^2 in the equation:
5u^2 - 4u√(1 - u^2) - 3 = 0.
Step 7: Solve the quadratic equation.
Now we can solve the quadratic equation using factoring, the quadratic formula, or any other suitable method. Once you find the values of u (sin(x)), you can solve for x by taking the inverse sine (arcsin) of those values.
Note: When solving the quadratic equation, you may obtain real solutions, complex solutions, or no solutions at all.
By following these steps, you can analyze the equation and solve for the unknown variable x.