What is the solution and exact values of 2cos^2Ø-7cosØ-4=0?

Ф = 120° and Ф = 240°

Step-by-step explanation:

Temporarily replace cos Ф with x, and solve the resulting quadratic equation for x:

2x^2 - 7x - 4 = 0.   This factors as follows:  (2x + 1)(x - 4) = 0, and so x = -1/2 and x = 4.

But we let x represent cos Ф.  Thus, we need to solve the two equations

x = cos Ф = -1/2 and x = cos Ф = 4.  The latter equation has no solution, as -1 ≤ cos Ф ≤ 1.  

cos Ф is negative in Quadrants II and III.  If in Quadrant II, the adjacent side is -1 and the hypotenuse is 2nd so Ф = arccos -1/2 = 120°.  If in Quadrant III, the adjacent side is -1 again and the hypotenuse 2 again, so Ф = 240°.

The exact solutions are Ф = 120° and Ф = 240°.

step-by-step explanation:

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