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Chapter 5: Problem 30

Factor the trigonometric expression. There is more than one correct form of each answer. $$6 \cos ^{2} x+5 \cos x-6$$

Short Answer

Expert verified
The factorized form of the given trigonometric expression is \((\cos x+1)(\cos x-1/2)\).

Step by step solution

01

Identify the quadratic form of the given expression

The given expression is \(6 \cos ^{2} x+5 \cos x-6\), we note that it is in quadratic form which is \(ax^2+bx+c\). Here, \(\cos x\) is 'x' and a=6, b=5, and c=-6.
02

Factorize the quadratic

The expression is now seen as a standard quadratic, recall that a quadratic \(ax^2+bx+c\) can be expressed as \((x-p)(x-q)\) where 'p' and 'q' are roots of the equation \(ax^2+bx+c = 0\). Solving \(6x^2+5x-6=0\) gives x=-1 and x=1/2. Therefore, our quadratic can be factored as \((x+1)(x-1/2)\)
03

Replace 'x' with \(\cos x\)

Substitute \(\cos x\) in place of x in the solution obtained in step 2, we get: \((\cos x+1)(\cos x-1/2)\).

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Most popular questions from this chapter

Use the product-to-sum formulas to rewrite the product as a sum or difference. $$\cos 2 \theta \cos 4 \theta$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\sin 6 x+\sin 2 x=0$$

(a) determine the quadrant in which \(u / 2\) lies, and (b) find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\tan u=-5 / 12, \quad 3 \pi / 2

Write the expression as the sine, cosine, or tangent of an angle. $$\cos 3 x \cos 2 y+\sin 3 x \sin 2 y$$

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\cos \left(\theta+\frac{\pi}{2}\right)-\cos \left(\theta-\frac{\pi}{2}\right)$$
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