Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$1-\cos ^{2} \theta$$

Step-by-Step Solution

Answer

The simplified expression is \(\sin^2 \theta\).

Step 1: Identify the Trigonometric Identity

Start by recognizing that the expression involves a known trigonometric identity. Here, we have \(1 - \cos^2 \theta\), which is related to the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1\).

Step 2: Apply the Pythagorean Identity

Use the Pythagorean identity to rewrite \(1 - \cos^2 \theta\) in terms of sine. According to the identity, \(\sin^2 \theta = 1 - \cos^2 \theta\). Substitute \(\sin^2 \theta\) for \(1 - \cos^2 \theta\).

Step 3: Simplify the Expression

After rewriting the expression using the identity, we get \(\sin^2 \theta\). Since this is already a simplified expression using basic trigonometric functions, our solution is \(\sin^2 \theta\).

Key Concepts

Sine and Cosine

Sine and cosine are fundamental concepts in trigonometry, representing two of the primary trigonometric functions. These functions are crucial because they relate the angles and sides of a right triangle. The sine function, denoted as \(\sin \theta\), is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Meanwhile, the cosine function, denoted as \(\cos \theta\), is the ratio of the length of the adjacent side to the hypotenuse.
Understanding these functions is essential because they form the basis for defining other trigonometric identities.
Practically, sine and cosine are used in various applications such as physics, engineering, and computer graphics to model periodic phenomena.

• Sine Function: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
• Cosine Function: \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)

When faced with expressions involving sine and cosine, think of them as components of a "complete picture" of the angle or the trigonometric problem you are trying to solve.

Pythagorean Identity

The Pythagorean identity is one of the most critical trigonometric identities, providing a relationship between sine and cosine. It states that for any angle \(\theta\), the sum of the squares of sine and cosine is always equal to one:
\[\sin^2 \theta + \cos^2 \theta = 1\]
This identity is derived from the Pythagorean theorem applied in a unit circle context. It becomes invaluable for simplifying trigonometric expressions and solving trigonometric equations.
When you encounter a term like \(1 - \cos^2 \theta\), it's beneficial to recognize it as \(\sin^2 \theta\) due to this identity.

• Uses: Simplifying expressions, solving trigonometric equations.
• Application: Finding sine given cosine or vice versa.

Recognizing and applying the Pythagorean identity simplifies problems and reduces the complexity of trigonometric expressions.

Expression Simplification

Expression simplification is the process of reducing mathematical expressions to their simplest form. In trigonometry, this often involves using identities like the Pythagorean identity to transform and simplify the expressions.
For the expression \(1 - \cos^2 \theta\), simplification involves recognizing it as \(\sin^2 \theta\) using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\).
Simplifying an expression is crucial because it makes equations easier to work with, recognizes relationships, and provides insights into trigonometric behavior in a given context.

• Steps: Identify the form, apply identities, reduce to simplest terms.
• Benefits: Easier problem solving, clearer understanding.

By reducing expressions to simpler forms, one can more easily perform calculations and make accurate predictions in trigonometry and applied mathematics.