Problem 58 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\cos (x+\pi)=-\cos x$$

Step-by-Step Solution

Answer

Short answer: By using the cosine angle addition formula, \(\cos (A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B\), we plugged in the values \(x\) and \(\pi\) for \(A\) and \(B\) respectively. After evaluating \(\cos \pi\) and \(\sin \pi\), we simplified the expression to find that \(\cos (x + \pi) = -\cos x\), proving the trigonometric identity.

Step 1: Recall the cosine angle addition formula

The cosine angle addition formula is: $$\cos (A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B$$ In this exercise, we have \(A = x\) and \(B = \pi\). So we need to plug these values into the formula.

Step 2: Apply the formula to the given identity

Replace \(A\) with \(x\) and \(B\) with \(\pi\) in the formula: $$\cos (x + \pi) = \cos x \cdot \cos \pi - \sin x \cdot \sin \pi$$

Step 3: Evaluate cos and sin values of known angles

Find the values of \(\cos \pi\) and \(\sin \pi\). Since \(\pi\) is found at the point \((-1,0)\) on the unit circle, the values are: $$\cos \pi = -1 \text{ and } \sin \pi = 0$$

Step 4: Simplify the expression

Substitute the values found in Step 3 back into the equation: $$\cos (x + \pi) = \cos x \cdot (-1) - \sin x \cdot 0$$ This simplifies to: $$\cos (x + \pi) = -\cos x$$

Step 5: Conclude the proof

The given identity has been proven: $$\cos (x + \pi) = -\cos x$$

Key Concepts

Cosine Addition Formula

The cosine addition formula is a vital tool in trigonometry that helps us understand how the cosine function behaves when adding two angles together. The general formula is given by:\[\cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B\]This formula essentially helps in breaking down the complex expression of \(\cos(A+B)\) into simpler terms using the cosines and sines of the angles \(A\) and \(B\). It uses the fundamental properties of these trigonometric functions.For example, in proving the identity \(\cos(x+\pi) = -\cos x\), we use this formula by substituting \(A\) with \(x\) and \(B\) with \(\pi\). This simplifies the problem by expressing \(\cos(x+\pi)\) in terms of \(\cos x\) and known trigonometric values like \(\cos \pi\) and \(\sin \pi\). As a result, it highlights the symmetry and periodicity properties of trigonometric functions. The cosine addition formula is widely used in various scenarios such as transforming trig equations, proving identities, and even solving complex triangles in geometry.

Unit Circle

The unit circle is a powerful concept in trigonometry that serves as a foundational tool for understanding trigonometric functions. The unit circle is a circle with a radius of one centered at the origin (0,0) on the coordinate plane. It is especially useful because it provides a geometric interpretation of the trigonometric functions' values at different angles.Each point on the unit circle corresponds to an angle \(\theta\) measured from the positive x-axis. The x-coordinate of this point is \(\cos \theta\), and the y-coordinate is \(\sin \theta\). For example, the angle \(\pi\) corresponds to the point (-1, 0) on the unit circle.

• \(\cos \pi = -1\)
• \(\sin \pi = 0\)

These coordinates are essential when evaluating expressions such as \(\cos(x+\pi)\) because they allow us to immediate plug in the known values derived from the unit circle onto the cosine addition formula.Ultimately, the unit circle not only aids in simplifying trigonometric calculations but also aids visualization of how trigonometric functions behave as the angle increases or decreases. It provides a consistent way to find the sine and cosine of any angle, making it a universal tool in trigonometry.

Trigonometric Functions

Trigonometric functions, including sine and cosine, are critical in understanding the relationships between angles and sides in right triangles, but their application extends far beyond triangles. These functions are periodic, oscillating in a wave-like pattern, and are expressed mathematically in terms of angles.

Key Trigonometric Functions:

• Cosine (\(\cos\)): Measures the horizontal distance relative to the angle on the unit circle.
• Sine (\(\sin\)): Measures the vertical distance relative to the angle on the unit circle.
• Tangent (\(\tan\)): A ratio of sine and cosine, given by \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

These functions are intrinsic to the study of waves, oscillations, and are utilized in various fields such as physics, engineering, and even economics. For instance, in the given identity proof, \(\cos(x+\pi)\) reflects on how trigonometric functions are not limited to angles in the initial quadrant only but extend to all angles, positive and negative, showcasing their periodic nature.Understanding these functions and their properties help us solve complex problems involving waves, vibrations, and alternating currents, among others. Trigonometric functions offer us a language to describe and predict cyclical patterns in both mathematical and real-world contexts.