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Chapter 5: Problem 74

Show that $$\cos (2 \theta) \leq \cos ^{2} \theta$$ for every angle \(\theta\).

Short Answer

Expert verified
Using the double-angle formula, \(\cos(2\theta) = 2\cos^2(\theta) - 1\), we rewrite the inequality as \(2\cos^2(\theta) - 1 \leq \cos^2(\theta)\). Rearranging and recognizing the trigonometric identity \((1 - \cos^2(\theta))^2\), we observe that this square is always non-negative, confirming that \(\cos(2\theta) \leq \cos^2(\theta)\) holds true for any angle 胃.

Step by step solution

01

Express \(cos(2\theta)\) using the double angle formula

We start by rewriting \(\cos(2\theta)\) using the double-angle formula, which states that \(\cos(2\theta) = 2\cos^2(\theta) - 1\).
02

Rewrite the inequality in terms of double angle and the original angle

Now, substitute this double-angle formula for \(\cos(2\theta)\) in the inequality: \(2\cos^2(\theta) - 1 \leq \cos^2(\theta)\)
03

Rearrange the inequality

Rearrange the inequality and take the left-hand side: \(\cos^2(\theta) - 2\cos^2(\theta) + 1 \geq 0\)
04

Identify a trigonometric identity

Identify the expression as a trigonometric identity: \((1 - \cos^2(\theta))^2 \geq 0\)
05

Observe that a square is always non-negative

As the square of any real number is always non-negative, it is evident that: \((1 - \cos^2(\theta))^2 \geq 0\) This confirms that the inequality \(\cos(2\theta) \leq \cos^2(\theta)\) holds true for any angle 胃.

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Key Concepts

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

Double Angle Formulas
The double angle formulas are a fundamental part of trigonometry, used to express trigonometric functions of doubled angles in terms of single angles.
One key formula is for the cosine function:
  • \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)
This formula helps to simplify expressions and solve equations where the angle is doubled.
It's derived from adding the angle sum identities, specifically for cosine.
By knowing the double angle formula, you can relate complex trigonometric functions back to simpler ones.
In problems like the given inequality, using this formula allows you to manipulate the expression and simplify the problem statement effectively.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the input angle.
They allow the transformation and simplification of expressions.
For this inequality, consider:
  • \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)
  • Basic identity: \( \sin^2\theta + \cos^2\theta = 1 \)
Using these identities, expressions can be rewritten to reveal underlying patterns.
In this case, manipulating \( 2\cos^2(\theta) - 1 \) into \( \cos^2(\theta) - 2\cos^2(\theta) + 1 \), highlights how identities can transform complex forms into simpler ones.
Cosine Function
The cosine function describes the adjacent side over the hypotenuse in a right angle triangle.
It's one of the core trigonometric functions, crucial for understanding the behavior of periodic phenomena.
  • \( \cos\theta \) ranges between -1 and 1.
  • At \( \theta = 0 \), \( \cos\theta = 1 \); at \( \theta = 90^\circ \, (\pi/2) \), \( \cos\theta = 0 \).
In the inequality \( \cos(2\theta) \leq \cos^2(\theta) \), the properties of the cosine function can be explored.
Understanding the range helps to visualize why \((1 - \cos^2(\theta))^2 \geq 0 \) holds true.
Exploring how the cosine function behaves invokes the intuition and graphical interpretation needed to approach trigonometric inequalities with confidence.

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Most popular questions from this chapter

The next two exercises emphasize that \(\cos (x+y)\) does not equal \(\cos x+\cos y\). For \(x=19^{\circ}\) and \(y=13^{\circ}\), evaluate each of the following: (a) \(\cos (x+y)\) (b) \(\cos x+\cos y\)

Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\sin (u-v)$$

Show that $$|\cos x+\sin x| \leq \sqrt{2}$$ for every number \(x\).

Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\tan (u-v)$$

Find the angle between a side of length 5 and the side with length 9 in an isosceles triangle that has one side of length 9 and two sides of length \(5 .\)
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