Problem 43 - Calculus Single Variable

Question

Use one or more of the basic trigonometric identities to derive the given identity. \(\cos (\theta+\pi)=-\cos (\theta)\)

Step-by-Step Solution

Answer

The identity \( \cos (\theta+\pi)=-\cos (\theta) \) is proven using the cosine addition formula.

Step 1: Recognize the Addition Formula for Cosine

The addition formula for cosine is given by: \[ \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b). \] Our expression involves \( \cos(\theta+\pi) \), so we will apply this formula by letting \( a = \theta \) and \( b = \pi \).

Step 2: Apply the Values for Trigonometric Functions at \( \pi \)

We know that \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). Using these, substitute into the addition formula: \[ \cos(\theta + \pi) = \cos(\theta)\cos(\pi) - \sin(\theta)\sin(\pi) = \cos(\theta)(-1) - \sin(\theta)(0). \]

Step 3: Simplify the Expression

The second term of the expression, \( \sin(\theta)(0) \), simplifies to 0. Hence, the expression simplifies to: \[ \cos(\theta + \pi) = -\cos(\theta). \] This completes the derivation and proves the given identity.

Key Concepts

Addition Formula for Cosine

The addition formula for cosine is a fundamental part of trigonometry. It allows us to calculate the cosine of a sum of two angles. Mathematically, it is expressed as:

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

This formula is useful because it breaks down the cosine of a composite angle into components that might be easier to calculate. It is particularly helpful when dealing with angles that are commonly found in the unit circle.
By applying the addition formula, we can simplify complex trigonometric expressions and prove identities effectively.
In our exercise, we identified \(a\) as \(\theta\) and \(b\) as \(\pi\), setting the stage for exploring trigonometric identities.

Cosine Function and Its Properties

The cosine function is one of the basic trigonometric functions. It is denoted as \(\cos(\theta)\), where \(\theta\) is the angle measured in radians or degrees. The function provides the x-coordinate of a point on the unit circle corresponding to a given angle.

• The period of \(\cos(\theta)\) is \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
• Cosine is an even function, which means \(\cos(-\theta) = \cos(\theta)\).

This property of symmetry is very useful in simplifying expressions and understanding trigonometric identities.
In the exercise, recognizing that \(\cos(\pi) = -1\) helps us apply the addition formula and understand the transformation of the cosine function when shifted by \(\pi\).

Understanding Trigonometric Functions

Trigonometric functions are essential in mathematics, describing the relationships between the angles and sides of triangles. They are also used to model periodic phenomena like waves and oscillations. The main trigonometric functions are sine, cosine, and tangent.

• \(\sin(\theta)\) and \(\cos(\theta)\) are based on coordinates of the unit circle.
• \(\tan(\theta)\) is the ratio \(\sin(\theta)/\cos(\theta)\).

Understanding these functions is key to mastering trigonometry.
They help derive identities and simplify complex angles by breaking them into parts that are easier to analyze. In our example, knowing the values of \(\sin(\pi)\) and \(\cos(\pi)\) were crucial to simplifying and verifying the given identity.

The Concept of Angle Addition

Angle addition is a concept that combines different angles to create composite angles. This is particularly useful in trigonometry when working with angle sums or differences.
By using the addition formulas for sine and cosine, we can manage complex angle expressions.

• The addition formula for cosine is \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\).
• Similarly, the sine addition formula is \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \).

These formulas allow us to express the trigonometric function of a sum in terms of the trigonometric functions of the individual angles.
Understanding angle addition helps in various applications, such as solving trigonometric equations, proving identities, and even in calculus. In our context, using the angle \(\theta + \pi\) showcases how angle addition can simplify and break down complex expressions for more straightforward analysis.