Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(180^{\circ}+x\right)$$

Step-by-Step Solution

Answer

\(-\cos(x)\)

Step 1: Identify the Trigonometric Identity

Recognize the identity used in the given expression. The expression \(\cos(180^{\circ} + x)\) can be transformed using the cosine addition identity. Specifically, use the identity \(\cos(180^{\circ} + x) = -\cos(x)\). This is due to the standard angle transformation for cosine.

Step 2: Apply the Identity

Apply the identity identified in the first step directly to the expression. Since \(180^{\circ}\) belongs to a quadrant where the cosine function changes sign, substitute the expression using the identity: \(\cos(180^{\circ} + x) = -\cos(x)\).

Step 3: Simplify the Expression

With the identity applied, the expression simplifies to just \(-\cos(x)\). There are no further terms or operations to consider, as \(-\cos(x)\) is already in its simplest form.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(\theta) \), is one of the primary trigonometric functions, along with sine and tangent. It relates primarily to the x-coordinate of the point on the unit circle where the terminal side of the angle \( \theta \) intersects. The cosine function outputs the cosine of an angle, which is the ratio of the adjacent side to the hypotenuse in a right triangle.

Important properties of cosine include:

• Even Function: Meaning \( \cos(-\theta) = \cos(\theta) \).
• Periodicity: Cosine has a period of \(360^{\circ}\) or \(2\pi\), so \( \cos(\theta + 360^{\circ}) = \cos(\theta) \).
• Range: The values of cosine are always between -1 and 1.

Cosine plays a crucial role in various mathematical and real-world applications, from engineering to wave mechanics. Understanding this function helps in the transformation and manipulation of trigonometric expressions.

Angle Transformation

Angle transformation refers to changing the representation of an angle. This is often necessary to simplify trigonometric expressions. For example, the expression \( \cos(180^{\circ} + x) \) uses the property of angles and trigonometric functions.

In this transformative approach:

• The angle \(180^{\circ} + x\) is significant because it's in the second quadrant of the unit circle.
• In the second quadrant, cosine values are negative, so applying the identity gives \(-\cos(x)\).
• Angle transformations help convert complex angles into simpler expressions.

These transformations make it easier to understand and manipulate trigonometric problems, leveraging symmetric properties of the trigonometric functions.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are foundational elements in mathematics for studying triangles, circles, and periodic phenomena. Each function relates an angle of a right triangle to ratios of two side lengths.

Some key characteristics of these functions include:

• Periodic Nature: Trigonometric functions repeat in regular patterns, a property that is very useful in modelling wave-like phenomena.
• Unit Circle Relation: Angles and trigonometric values can be found using their positions on the unit circle, where the radius is 1.
• Complementary Relationships: The functions have relationships such as \( \sin(\theta) = \cos(90^{\circ} - \theta) \).

Understanding these functions allows for solving equations, graphing periodic functions, and transforming angles efficiently, crucial for higher-level mathematics and practical applications.