Problem 113 - Precalculus : Building Concepts and Connections

Question

Suppose \(t\) is in \(\left(0, \frac{\pi}{2}\right) .\) Express \(\cos (t+\pi)\) in terms of cos \(t .\) (Hint: It is helpful to sketch a figure.)

Step-by-Step Solution

Answer

The expression \( \cos(t+\pi) \) expressed in terms of \( \cos(t) \) is \( -\cos(t) \)

Step 1: Identify the required trigonometric identity

The main identity involved in the exercise is the cosecant angle sum identity. It's expressed as \( \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \). Here, \(a = t\) and \(b = \pi\)

Step 2: Application of the the cosine angle sum identity

By applying the identity to \( \cos(t+\pi) \), we get: \( \cos(t+\pi) = \cos(t)\cos(\pi) - \sin(t)\sin(\pi) \)

Step 3: Utilize the characteristics of sine and cosine functions

We know that \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). Substituting these values into the previous expression, we get: \( \cos(t+\pi) = \cos(t)*(-1) - \sin(t)*0 \)

Step 4: Simplify the expression

Since any number multiplied by zero is zero, the second term drops out. The remaining expression is \( \cos(t+\pi) = -\cos(t) \)

Key Concepts

Cosine Angle Sum Identity

The cosine angle sum identity is a cornerstone of trigonometry, providing a way to express the cosine of the sum of two angles in terms of the sines and cosines of the individual angles. The identity states that for any two angles, denoted here as \( a \) and \( b \), the cosine of their sum is given by the equation \( \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \).

This identity is particularly useful when simplifying expressions or solving equations involving trigonometric functions. In the context of our exercise, we apply the cosine angle sum identity to determine \( \cos(t+\pi) \). By recognizing that \( a = t \) and \( b = \pi \), we substitute these values into the identity, leading us to the simplified result \( \cos(t+\pi) = -\cos(t) \), which demonstrates the periodic properties of cosine.

Properties of Sine and Cosine Functions

The sine and cosine functions are two of the most fundamental trigonometric functions. They possess unique properties that are critical to understand when studying trigonometry. Among these properties are their values at key angles, their periodicity, and their even and odd nature.

• Key Angles: For example, \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \), which we use to simplify the cosine angle sum expression in our problem.
• Periodicity: Both sine and cosine functions have a period of \( 2\pi \), meaning they repeat their values every \( 2\pi \) radians.
• Even and Odd Nature: Cosine is an even function, so \( \cos(-x) = \cos(x) \), whereas sine is an odd function, meaning \( \sin(-x) = -\sin(x) \).

These properties allow us to manipulate trigonometric expressions effectively, as can be seen when simplifying \( \cos(t+\pi) \) into \( -\cos(t) \) in our example.

Trigonometric Functions Transformations

Understanding trigonometric functions also involves examining how they transform when subjected to shifts, stretches, and reflections. In trigonometry, transformations can change the amplitude, period, phase, or orientation of the function's graph without altering its shape.

For example, adding \( \pi \) to the argument of the cosine function in the expression \( \cos(t+\pi) \) effectively reflects the graph of \( \cos(t) \) across the y-axis, leading to a phase shift that significantly alters the function's output. Such transformations are crucial for both understanding the behavior of trigonometric functions and solving complex equations involving them.

In our exercise, these transformative properties explain why adding \( \pi \) to the angle \( t \) inverts the sign of the cosine function, which is a specific type of transformation known as a horizontal shift. Identifying and effectively using these transformations can simplify the process of working with trigonometric functions in various applications.