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Chapter 5: Problem 17

Use a double-angle formula to rewrite the expression. \(6 \cos ^{2} x-3\)

Short Answer

Expert verified
The rewritten expression is \(3\cos 2x\).

Step by step solution

01

Identify the appropriate double-angle identity

The given expression looks similar to the double-angle identity \(\cos 2x = 2 \cos^{2}x - 1\). It can be rewritten in terms of \(\cos 2x\) if we manipulate it appropriately.
02

Rewrite the given expression

The given expression is \(6 \cos ^{2} x-3\). We can factor out 3 to give us \(3(2\cos^{2}x - 1)\).
03

Apply the Double Angle Identity

Replace the \(2\cos^{2}x - 1\) part of the expression with \(\cos 2x\). This will give us the final result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine
Cosine is one of the primary trigonometric functions. It measures the horizontal coordinate or the x-component of a point on the unit circle, given a specific angle. Particularly, in right-angled triangles, it represents the ratio of the adjacent side to the hypotenuse. Cosine is valuable because of its various properties and ratios:
  • Cosine has a range from -1 to 1.
  • It is an even function, meaning \(\cos(-x) = \cos(x)\) for any angle x.
  • It's periodic, repeating every 360掳 or \(2\pi\) radians.
Cosine functions are fundamental to trigonometry and help solve more complex geometric and analytical problems.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They are critical tools for simplifying expressions and solving trigonometric equations. The common identities include:
  • Pythagorean identities
  • Angle sum and difference identities
  • Double-angle identities
In our context, the double-angle identities relate angles that are twice another angle. For cosine, this identity is expressed as:\[\cos(2x) = 2\cos^{2}(x) - 1\]This identity allows us to express cosine of a double angle in terms of the square of a single cosine value, which simplifies calculations and expression manipulations.
Precalculus
Precalculus lays the groundwork for calculus, but it's more than just an introduction. It covers essential algebraic and trigonometric concepts needed to advance in mathematical studies. Key areas in precalculus include:
  • Functions and their properties
  • Trigonometry and the unit circle
  • Polynomials and rational functions
A significant focus is placed on understanding and manipulating expressions using trigonometric identities, such as the double-angle formulas. These concepts are vital for calculus, where they appear frequently in differential and integral calculus.
Trigonometric Functions
Trigonometric functions describe the relationship between angles and lengths in triangles. They are foundational in mathematics, especially in areas involving periodic phenomena, like waves and oscillations. The main trigonometric functions include sine, cosine, and tangent, each with a distinct role:
  • Sine measures the y-component or vertical distance on the unit circle.
  • Cosine measures the x-component or horizontal distance.
  • Tangent is the ratio of sine to cosine.
These functions are extended to the entire unit circle, making them applicable beyond right triangles, thus critical in various mathematical and physical applications.

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Most popular questions from this chapter

Prove the identity.$$\sin \left(\frac{\pi}{2}+x\right)=\cos x$$

Find the exact value of the trigonometric expression given that \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} .\) (Both \(u\) and \(v\) are in Quadrant II.)$$\sec (v-u)$$.

Use the figure, which shows two lines whose equations are \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\). Assume that both lines have positive slopes. Derive a formula for the angle between the two lines. Then use your formula to find the angle between the given pair of lines.\(y=x\) and \(y=\frac{1}{\sqrt{3}} x\).

Find the exact value of the trigonometric expression given that \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} .\) (Both \(u\) and \(v\) are in Quadrant II.)$$\cos (u-v)$$ .

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\).$$\cos \left(x-\frac{\pi}{2}\right)-\sin ^{2} x=0$$
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