Problem 40 - Contemporary Precalculus

Question

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\cos (-t)=\cos t$$

Step-by-Step Solution

Answer

Answer: Yes, the equation $$\cos(-t) = \cos(t)$$ is an identity, as the graphs of the two functions coincide and the cosine function is known to be an even function.

Step 1: Recall the properties of cosine function

The cosine function is an even function, which means that for any angle 't', we have $$\cos(-t) = \cos(t)$$. Keep this property in mind when analyzing the given equation.

Step 2: Graph $$y_1 = \cos(t)$$ and $$y_2 = \cos(-t)$$

Using a graphing tool or software, plot the graphs of the functions $$y_1 = \cos(t)$$ and $$y_2 = \cos(-t)$$ for a range of 't' values, such as from $$-2\pi$$ to $$2\pi$$.

Step 3: Compare the graphs

Observe the graphs of both functions $$y_1 = \cos(t)$$ and $$y_2 = \cos(-t)$$. If they coincide (overlap exactly), then the equation $$\cos(-t) = \cos(t)$$ is an identity. This is because the cosine function is even, and as discussed in Step 1, the properties of even functions reflect this equality.

Step 4: Conclusion

Since the graphs of $$y_1 = \cos(t)$$ and $$y_2 = \cos(-t)$$ coincide, the equation $$\cos(-t) = \cos(t)$$ is an identity. This confirms the known property that the cosine function is an even function.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry, primarily because of its role in defining relationships in a right triangle and its application in the unit circle. One of the key characteristics of the cosine function is that it is an **even function**. This means that for any angle \( t \), the function satisfies the identity \( \cos(-t) = \cos(t) \).

Even functions have a unique symmetry property where the graph is symmetric about the y-axis. In simpler terms, if you were to fold the graph along the y-axis, both halves would match perfectly. This property of the cosine function is extremely useful and explains the consistent mirror effect around the y-axis, making it a vital concept when solving trigonometric equations and identities.

Graphing Trigonometric Functions

Graphing trigonometric functions such as the cosine function helps us visualize and understand their behavior and properties. To graph \( y = \cos(t) \) and \( y = \cos(-t) \), you would observe that both graphs overlap completely. This overlap visualizes the even nature of the cosine function.

When graphing trigonometric functions, it's important to consider the range of values for the angle \( t \). For cosine, a common range is from \( -2\pi \) to \( 2\pi \), covering one full cycle. The cosine function’s graph is a repeating wave-like curve that reaches its maximum value of 1 and minimum value of -1.

• **Amplitude:** The height from the centerline to the peak of the wave, which for cosine is 1.
• **Period:** The distance over which the wave repeats, which is \( 2\pi \).
• **Phase Shift:** Determines the horizontal shifting of the wave. For the basic cosine function, there is no phase shift.

Understanding and being able to graph these properties help in analyzing trigonometric expressions and identifying identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value in the domain. In the context of cosine, one of the simplest but most crucial identities is \( \cos(-t) = \cos(t) \). This identity results from the even property of the cosine function.

Trigonometric identities serve a significant function in simplifying complex expressions, proving equivalences, and solving trigonometric equations. Some well-known trigonometric identities include:

• **Pythagorean Identities:** Such as \( \cos^2(t) + \sin^2(t) = 1 \).
• **Angle Sum Identities:** Like \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \).
• **Double Angle Identities:** Such as \( \cos(2t) = \cos^2(t) - \sin^2(t) \).

These identities are tools that help trigonometry students solve problems across various mathematical disciplines and applications in physics, engineering, and beyond.