Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos (\pi-x)$$

Step-by-Step Solution

Answer

\( \cos(\pi - x) = -\cos(x) \)

Step 1: Understanding the Problem

We need to express \( \cos (\pi - x) \) as a function with \( x \) as the only argument. This involves recognizing an identity that applies to this expression.

Step 2: Recall the Cosine Difference Identity

The cosine of a difference identity is \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). Here, \( a = \pi \) and \( b = x \).

Step 3: Evaluate the Identity Components

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

Step 4: Simplify Using Known Values

We know \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). Substitute these values to get \( \cos(\pi - x) = (-1)\cos(x) + 0 \cdot \sin(x) \).

Step 5: Final Simplification

Simplify the expression: \( \cos(\pi - x) = -\cos(x) \). This is now expressed as a function of \( x \) alone.

Key Concepts

Cosine Function

The cosine function is one of the primary functions in trigonometry. It's often abbreviated as "cos." The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. When dealing with angles on the unit circle, the cosine of an angle gives you the x-coordinate of a point on the circle.
Knowing how to work with the cosine function is crucial in precalculus, as it helps solve various trigonometric problems. The symbolism of this function can be noted as:

• \( \cos \theta = \frac{ \text{Adjacent} }{ \text{Hypotenuse} } \)
• In the unit circle, \( \cos \theta \) is simply the x-coordinate of the point corresponding to angle \( \theta \).

Understanding cosine in various situations, like acute angles or angles in the unit circle, helps students solve many different types of mathematical problems. Knowing its properties, like periodicity and symmetry, is also very helpful in understanding further concepts in calculus.

Angle Subtraction

Trigonometric identities are often used to simplify expressions involving angles. One key identity is the angle subtraction identity. This identity is particularly useful when simplifying expressions that involve the cosine of a difference of two angles, such as \( \cos(a - b) \). The identity states that:

• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

This expression shows how the function combines the individual cosines and sines of two angles \(a\) and \(b\). In practical problems, you substitute known values into this formula, simplifying complex trigonometric expressions.
For instance, when you have \( \pi - x \) in an expression, substituting using this identity allows you to express it solely in terms of \(x\). This is crucial for solving many precalculus problems.

Precalculus

Precalculus serves as the foundation for calculus and involves the study of functions, identities, and other mathematical concepts that are pivotal for understanding calculus. One of these areas is trigonometry, which includes the study of sine, cosine, and tangent functions, along with their identities. Precalculus ensures that students are equipped with the necessary skills to manipulate and simplify expressions – an essential skill when transitioning to calculus.
Students learn to work with fundamental trigonometric identities, such as the Pythagorean identity, angle addition, and subtraction identities. These identities form the backbone of solving precalculus problems.

• Emphasis is placed on enhancing algebraic skills.
• Students often encounter lessons on manipulating equations, graphing functions, and understanding their transformations.

All these concepts and skills add up to provide the full foundational framework upon which calculus is built.