find in degrees the value of Ө in the interval O¡ÜӨ¡Ü 360 for which 2cos^2Ө-cosӨ-1=sin^2Ө
give your answers to 1 decimal place where appropriate
To solve the equation 2cos^2Ө - cosӨ - 1 = sin^2Ө, we can use basic trigonometric identities to simplify the equation. Here's how you can do it step by step:
First, let's convert the equation from using cos^2Ө to sin^2Ө using the identity: sin^2Ө = 1 - cos^2Ө.
So, we have:
2cos^2Ө - cosӨ - 1 = 1 - cos^2Ө
Let's simplify further:
2cos^2Ө - cosӨ - 1 = 1 - cos^2Ө
2cos^2Ө + cos^2Ө - cosӨ - 1 = 1
3cos^2Ө - cosӨ - 1 = 1
Now, let's move all terms to one side of the equation:
3cos^2Ө - cosӨ - 1 - 1 = 0
3cos^2Ө - cosӨ - 2 = 0
Next, let's factorize the quadratic equation:
(3cosӨ + 2)(cosӨ - 1) = 0
From here, we have two possible cases:
Case 1: 3cosӨ + 2 = 0
Solving this equation, we have:
3cosӨ = -2
cosӨ = -2/3
To find the value of Ө in degrees, we can use the inverse cosine function (cos^-1 on most scientific calculators) with the obtained value:
Ө = cos^-1(-2/3)
Now, let's evaluate this using a calculator:
Ө ≈ 143.1°
Case 2: cosӨ - 1 = 0
Solving this equation, we have:
cosӨ = 1
Again, using the inverse cosine function, we get:
Ө = cos^-1(1)
Ө ≈ 0°
So, the values of Ө, in degrees, that satisfy the equation in the given interval are approximately 143.1° and 0°.