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Chapter 10: Problem 106

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Show that \(\cos ^{3} x=\cos x-\sin ^{2} x \cos x\).

Short Answer

Expert verified
\cos^{3}x = \cos x \ - \ \sin^{2} x \cos x.

Step by step solution

01

Start with the given identity

We need to show that \ \ \ \ \ \ \ \( \cos ^{3} x=\cos x-\sin ^{2} x \cos x \).
02

Break down the right side

Rewrite the right side of the equation: \ \ \ \ \ \ \ \( \cos x-\sin ^{2} x \cos x \ =\cos x(1-\sin ^{2} x) \).
03

Simplify using Pythagorean Identity

Recall the Pythagorean Identity: \ \ \ \ \ \ \ \( 1 =\sin^{2} x +\cos^{2} x \). Substitute \( 1 -\sin^{2} x \) with \( \cos^{2} x. \Then, \cos x(1-\sin ^{2} x) =\cos x (\cos ^{2} x). \)
04

Verify the final expression

Multiply \ \ \ \ \cos x \ times \ \cos^{2} x, resulting in \ \ \ \( \cos^{3} x = \cos x \cdot \cos^2 x. \). This confirms the original given identity.

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Key Concepts

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

Pythagorean Identity
The Pythagorean Identity is one of the fundamental trigonometric identities. It states that for any angle \(x\), the relationship between the sine and cosine functions is described by:
\[ \text {sin}^2 x + \text {cos}^2 x = 1 \] This means that no matter the value of \(x\), the sum of the squares of the sine and cosine functions will always equal 1.

In our exercise, we used this identity to simplify the given equation. By expressing \(1 - \text {sin}^2 x\) in terms of \( \text {cos}^2 x\), we made it easier to show that \( \text {cos}^3 x = \text {cos} x - \text {sin}^2 x \text {cos} x \).
Trigonometric Simplification
Trigonometric simplification is the process of reducing trigonometric expressions to simpler forms. This often involves using fundamental identities like the Pythagorean Identity.

In our example, we simplified \( \text {cos} x - \text {sin}^2 x \text {cos} x\) by factoring out \( \text {cos} x\), giving us:
\( \text {cos} x(1 - \text {sin}^2 x) \).

Next, we used the Pythagorean Identity to replace \(1 - \text {sin}^2 x\) with \( \text {cos}^2 x\), turning our expression into:
\(\text {cos} x \text {cos}^2 x)\).
This simplifies to \( \text {cos}^3 x\), matching the given identity.
Cosine Function
The cosine function, often denoted as \( \text {cos} x \), is one of the principal trigonometric functions. It represents the adjacent side over the hypotenuse in a right-angled triangle.

The cosine function also has several important properties and identities linked to it. For instance, \( \text {cos}^2 x = 1 - \text {sin}^2 x\), which is fundamental to many trigonometric simplifications.

In our problem, understanding the cosine function and its properties helped us simplify \( \text {cos} x - \text {sin}^2 x \text {cos} x\) to \( \text {cos} ^{3} x \). Using properties like these makes solving trigonometric identities more straightforward.

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Most popular questions from this chapter

Given vectors \(\mathbf{u}=2 x \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{v}=x \mathbf{i}-8 \mathbf{j},\) find \(x\) so that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-8 \mathbf{j} $$

In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive \(x\) -axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees. (c) Guaranteed Rate Field, home of the White \(\operatorname{Sox},\) is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of Guaranteed Rate Field using rectangular coordinates. (d) Write the location of Guaranteed Rate Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees.

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of \(5 \cos 60^{\circ}+2 \tan \frac{\pi}{4} .\) Do not use a calculator.

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