Problem 67 - Precalculus: A Concise Course

Question

Verify that \(\cos 2 t \neq 2 \cos t\) by approximating \(\cos 1.5\) and \(2 \cos 0.75\)

Step-by-Step Solution

Answer

Upon calculating, \(\cos(1.5)\) approximately equals 0.070737 and \(2 \cos(0.75)\) approximately equals 1.050643. As these two values are not close, we can confirm that \(\cos(2t)\) \(\neq\) \(2 \cos(t)\) for the given values of \(t\).

Step 1: Calculate \(\cos(1.5)\)

We first calculate \(\cos(1.5)\) using a calculator (or python function). This will give us the value for \(\cos(2t)\) where \(t=0.75\).

Step 2: Calculate \(2 \cos(0.75)\)

Next, we calculate \(2 \cos(0.75)\) using a calculator (or python function). This will give us the value for \(2 \cos(t)\) where \(t=0.75\).

Step 3: Compare the results

Finally, we compare the two results obtained in steps 1 and 2. If they are not equal, it verifies that \(\cos(2t)\) is not equal to \(2 \cos(t)\).

Key Concepts

Cosine Function

The cosine function is one of the primary functions in trigonometry. It is especially useful in representing the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
It is generally expressed as \( \cos(\theta) \), where \( \theta \) is the angle in question.
Employing cosine in trigonometric functions aids in calculating wave motions, circular motion, and periodic phenomena.
Understanding the cosine function is crucial to solving trigonometric identities, allowing for comparison and verification of different equations.

• In the unit circle, the x-coordinate of a point on the circle corresponds to the cosine of the angle.
• The cosine function has a range of -1 to 1, and it is periodic with a period of \(2\pi\).
• It is an even function, which means \(\cos(-\theta) = \cos(\theta)\).

Double Angle Formula

The double angle formulae are powerful tools in trigonometry that simplify complex expressions and solve equations involving double angles.
For the cosine function, the double angle formula is given by:
\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \]This formula can also be expressed using only cosine or sine:

• \(\cos(2\theta) = 2\cos^2(\theta) - 1\)
• \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)

Utilizing the double angle formula allows us to transform \( \cos(2t) \) into terms of \( \cos(t) \) and \( \sin(t) \). This is essential in verifying identities like the one in the given exercise. As seen, \(\cos(2t) eq 2 \cos(t)\), illustrating the utility of the formula in checking trigonometric relationships.

Trigonometric Approximation

Trigonometric approximation comes into play when we need to estimate the values of trigonometric functions without analytic calculation.
Using a calculator or computational method, we can achieve precise estimates of functions like \( \cos(1.5) \) or \( 2\cos(0.75) \).
The purpose of approximation is crucial when exact values are impractical or impossible to obtain using basic identities. In this exercise:

• Approximating \( \cos(1.5) \) provides an estimate for \( \cos(2t) \).
• Approximating \( 2\cos(0.75) \) gives an estimate for \( 2\cos(t) \).

This approach confirms that the two values obtained are not equal, reinforcing that \( \cos(2t) eq 2\cos(t) \). Approximations are thus an effective tool for verifying and illustrating trigonometric principles in practical scenarios.