Problem 28 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\cos (x-\pi)=-\cos x$$

Step-by-Step Solution

Answer

The identity \( \cos(x - \pi) = -\cos x \) is proven using the cosine difference formula.

Step 1: Understand the Identity to Prove

The identity given is \( \cos(x - \pi) = -\cos x \). This means that the cosine of \(x\) shifted by \(\pi\) radians is the negative cosine value of \(x\).

Step 2: Use the Cosine Difference Formula

The cosine of a difference is given by the formula: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). Here, \( a = x \) and \( b = \pi \), so we apply the formula to get \( \cos(x - \pi) = \cos x \cos \pi + \sin x \sin \pi \).

Step 3: Evaluate Trigonometric Values at \( \pi \)

Now we substitute the known values of cosine and sine at \( \pi \): \( \cos \pi = -1 \) and \( \sin \pi = 0 \). This gives us \( \cos(x - \pi) = \cos x (-1) + \sin x (0) \).

Step 4: Simplify the Expression

Simplifying the expression from Step 3, we get: \( \cos(x - \pi) = -\cos x \cdot 1 + 0 = -\cos x \).

Step 5: Conclusion of the Proof

The simplified result matches the right side of the identity: \( \cos(x - \pi) = -\cos x \). Thus, the identity is proven as \( \cos(x - \pi) = -\cos x \).

Key Concepts

Cosine Function

The cosine function, often denoted as \( \cos \), is a fundamental trigonometric function. It is defined in the context of a right-angled triangle as the ratio of the length of the adjacent side to the hypotenuse. The cosine function is periodic with a period of \( 2\pi \), meaning that the function repeats its values every \( 2\pi \) radians.

In the unit circle representation, the cosine of an angle is the horizontal coordinate of the point determined by the angle from the origin. Due to the periodic nature, cosine values repeat at specific interval shifts like \( 2\pi \), but also exhibit symmetry properties. This includes an important characteristic shown in identities such as \( \cos(x - \pi) = -\cos x \).

• Cosine of 0° or 0 radians is 1.
• Cosine of 90° or \( \frac{\pi}{2} \) radians is 0.
• Cosine of 180° or \( \pi \) radians is -1.

Understanding these basic values aids in evaluating and understanding trigonometric expressions.

Angle Difference

The angle difference refers to the difference between two angles in trigonometric calculations. A powerful identity related to this is the cosine difference identity, which is expressed as \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). This formula helps simplify problems where angles are subtracted from each other.

In the given problem, this identity is used to solve \( \cos(x - \pi) \). By applying the formula:

• Set \( a = x \) and \( b = \pi \).
• Substitute into the cosine difference identity: \( \cos(x - \pi) = \cos x \cos \pi + \sin x \sin \pi \).

This approach helps show how shifts of angle by \( \pi \) radians result in a sign change in the trigonometric function.

Radians

Radians are a measurement for angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius. This is a natural way of measuring angles and is often preferred in mathematics due to its simplicity in calculus.

In the circle, \( 360^{\circ} \) corresponds to \( 2\pi \) radians. Here are some key radian values:

• Right angle \( (90^{\circ}) = \frac{\pi}{2} \) radians
• Straight line \( (180^{\circ}) = \pi \) radians
• Full circle \( (360^{\circ}) = 2\pi \) radians

Radians make it easier to compute trigonometric functions without conversions. In the context of our identity \( \cos(x - \pi) = -\cos x \), understanding \( \pi \) in radians helps explain why this shift causes cosine to become negative. As \( \pi \) radians describe half the circle, it effectively inverts the direction or phase of trigonometric functions.