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Chapter 7: Problem 30

Express each function as a trigonometric function of \(x .\) $$\cos 3 x$$

Short Answer

Expert verified
\text{cos}(3x) = 4 \text{cos}^3(x) - 3 \text{cos}(x)

Step by step solution

01

Use the triple angle formula for cosine

The triple angle formula for cosine is given by \[\begin{aligned} \ \text{cos}(3x) = 4 \text{cos}^3(x) - 3 \text{cos}(x)\end{aligned}\]

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Key Concepts

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

Cosine Triple Angle Formula
The cosine triple angle formula is a fundamental identity in trigonometry that simplifies the expression \(\text{cos}(3x)\).
It relates the cosine of a triple angle to the cosine of the original angle. The formula is given by: \[ \text{cos}(3x) = 4 \text{cos}^3(x) - 3 \text{cos}(x) \]
This means that to find \(\text{cos}(3x)\), you need to know the value of \(\text{cos}(x)\).
Here鈥檚 the formula in simpler words: the cosine of three times an angle is equal to four times the cube of the cosine of that angle, minus three times the cosine of the angle itself.
Cosine Function
The cosine function is one of the basic trigonometric functions.
It is often written as \(\text{cos}\).
It鈥檚 used to find the horizontal component of an angle. For any angle \(\theta\) in a right triangle, the cosine function is defined as the ratio of the adjacent side to the hypotenuse: \[ \text{cos}(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \] The cosine function has some important properties:
  • It is periodic, with a period of \(\text{2}\text{蟺}\). This means the function repeats its values every \(\text{2}\text{蟺}\) units.
  • The range of the cosine function is from -1 to 1.
  • It is an even function, which means \(\text{cos}(-x) = \text{cos}(x)\).
Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables involved.
They are useful to simplify trigonometric expressions and solve equations.
Some common trigonometric identities include:
  • Pythagorean Identities: \( \text{cos}^2(x) + \text{sin}^2(x) = 1 \)
  • Reciprocal Identities: \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\)
  • Angle Sum and Difference Identities: \[ \text{cos}(a \text{卤} b) = \text{cos}(a)\text{cos}(b) \text{\text{鈭搣} \text{sin}(a) \text{sin}(b) \]
Trigonometric identities are key tools in simplifying complex trigonometric problems.
Knowing these identities helps in solving equations and proving other trigonometric formulas.

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin 2 x=2 \cos ^{2} x$$

Verify that each trigonometric equation is an identity. $$\frac{1-\cos x}{1+\cos x}=\csc ^{2} x-2 \csc x \cot x+\cot ^{2} x$$

Use the given information to find \(\cos (s+t)\) and \(\cos (s-t)\). \(\sin s=\frac{2}{3}\) and \(\sin t=-\frac{1}{3}, s\) in quadrant II and \(t\) in quadrant IV

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. $$\sin \left(\frac{\pi}{2}+\theta\right)$$

Work each problem. Oscillating Spring The distance or displacement \(y\) of a weight attached to an oscillating spring from its natural position is modeled by $$ y=4 \cos (2 \pi t) $$ where \(t\) is time in seconds. Potential energy is the energy of position and is given by $$ P=k y^{2} $$ where \(k\) is a constant. The weight has the greatest potential energy when the spring is stretched the most. (Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol. \(2,\) Allyn \& Bacon.) (a) Write an expression for \(P\) that involves the cosine function. (b) Use a fundamental identity to write \(P\) in terms of \(\sin (2 \pi t)\) (figure cannot copy)
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