`2 cos^2 x - 7 cos x + 3 =0`
Ok consider the trigonometric equation as a quadratic expression.
Ok then set each factor equal to zero
`(2cosx-1)=0` and `(cosx-3)=0`
To solve for the angle x, ignore the factor cos x=3.Please note that the Range of Cosine function is from -1 to 1 only.
Hence consider only the first factor.
To determine the value of x, refer to Unit Circle Chart or Table of Trigonometeric Function for Special Angles.
x=60 deg and 300 deg
Since there is no indicated interval for the angle, then the general solution is
`x_1 =60+k(360) ` degress
`x_2 =300+k(360) ` degress
You should convert the given trigonometric equation into a quadratic equation, using the substitution `cos x = y` , such that:
`2y^2 - 7y + 3 = 0`
Using quadratic formula, yields:
`y_(1,2) = (7+-sqrt(49 - 24))/4`
`y_(1,2) = (7+-sqrt25)/4 => y_(1,2) = (7+-5)/4 => y_1 = 3; y_2 = 1/2`
You need to replace back cos x for y such that:
`cos x = 3` invalid since the values of cosine function cannot be larger than 1
`cos x = 1/2`
Since the cosine values are positive in quadrants 1 and 4 yields:
`x = pi/3` (quadrant 1)
`x = 2pi - pi/3 => x = (5pi)/3` (quadrant 4)
Hence, evaluating the exactly special angles that make the equation to hold, yields `x = pi/3, x = (5pi)/3.`