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Chapter 8: Problem 50

Find an expression for \(\cos (5 \theta)\) as a fifth-degree polynomial in the variable \(\cos \theta\).

Short Answer

Expert verified
Use the formula: \[ \cos(5\theta) = 16 \cos^5(\theta) - 20 \cos^3(\theta) + 5 \cos(\theta) \]

Step by step solution

01

- Use the Multiple Angle Formula

Recall the multiple angle formula for cosine: \[ \cos(5\theta) = 16 \cos^5(\theta) - 20 \cos^3(\theta) + 5 \cos(\theta) \]

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Key Concepts

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

Understanding the Multiple Angle Formula
The multiple angle formula is an essential tool in trigonometry. It helps to represent trigonometric functions of multiple angles in terms of single-angle expressions. For instance, \(\cos (5 \theta)\) can be expressed using the multiple angle formula: \[ \cos(5\theta) = 16 \cos^5(\theta) - 20 \cos^3(\theta) + 5 \cos(\theta) \] This means that the cosine of five times an angle (\(5\theta\)) can be broken down into a polynomial expression involving \(\cos(\theta)\). This simplification is useful for calculations and solving complex trigonometric problems. It is also a method to see patterns that can suggest further solution techniques.
The Role of Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all angles. These identities make it easier to manipulate and simplify trigonometric expressions. The multiple angle formula itself is a type of trigonometric identity that connects the angles and their trigonometric function values. Other common trigonometric identities include Pythagorean identities such as \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] and angle sum and difference identities like \[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\] These identities can be used in conjunction with the multiple angle formula to simplify equations and solve trigonometric problems in various contexts.
Link to Polynomial Expressions
The multiple angle formula for \(\cos (5 \theta)\) provides a link to polynomial expressions. Rewriting \(\cos (5 \theta)\) as \[ \cos(5\theta) = 16 \cos^5(\theta) - 20 \cos^3(\theta) + 5 \cos(\theta) \] shows how trigonometric functions can be represented as polynomials. This polynomial is of degree five since the highest power of \(\cos(\theta)\) is five. Polynomial expressions like this one are useful because they allow us to apply algebraic techniques to solve and analyze trigonometric problems. Additionally, understanding these polynomial relationships provides deeper insights into the behavior of trigonometric functions and their applications. They show how different areas of mathematics connect and can be used together effectively.

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Most popular questions from this chapter

Use the following discussion. The formula $$ D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right] $$ Approximate the number of hours of daylight for any location that is \(66^{\circ} 30^{\prime}\) north latitude for the following dates: (a) Summer solstice \(\left(i=23.5^{\circ}\right)\) (b) Vernal equinox \(\left(i=0^{\circ}\right)\) (c) July \(4\left(i=22^{\circ} 48^{\prime}\right)\) (d) Thanks to the symmetry of the orbital path of Earth around the Sun, the number of hours of daylight on the winter solstice may be found by computing the number of hours of daylight on the summer solstice and subtracting this result from 24 hours. Compute the number of hours of daylight for this location on the winter solstice. What do you conclude about daylight for a location at \(66^{\circ} 30^{\prime}\) north latitude?

Calculus Show that the difference quotient for \(f(x)=\cos x\) is given by $$ \begin{aligned} \frac{f(x+h)-f(x)}{h} &=\frac{\cos (x+h)-\cos x}{h} \\ &=-\sin x \cdot \frac{\sin h}{h}-\cos x \cdot \frac{1-\cos h}{h} \end{aligned} $$

One, Two, Three (a) Show that \(\tan \left(\tan ^{-1} 1+\tan ^{-1} 2+\tan ^{-1} 3\right)=0\). (b) Conclude from part (a) that $$ \tan ^{-1} 1+\tan ^{-1} 2+\tan ^{-1} 3=\pi $$

Find the exact value of each expression. $$ \cos \left(\tan ^{-1} \frac{4}{3}+\cos ^{-1} \frac{5}{13}\right) $$

A beam of light with a wavelength of 589 nanometers traveling in air makes an angle of incidence of \(40^{\circ}\) on a slab of transparent material, and the refracted beam makes an angle of refraction of \(26^{\circ} .\) Find the index of refraction of the material. \({ }^{t}\)
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