Problem 10 - Algebra 2

Question

Simplify each trigonometric expression. $$ 1-\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The simplified form of the trigonometric expression \(1 - \cos^2(θ)\) is \(\sin^2(θ)\).

Step 1: Recognize the Pythagorean Identity

Start by recognizing the Pythagorean identity. In trigonometry, the fundamental Pythagorean identity is \(\sin^2(θ) + \cos^2(θ) = 1\). It means that the square of the sine of an angle plus the square of the cosine of an angle is always equal to 1.

Step 2: Substitute the expression with Pythagorean Identity

Now, it's time to substitute the expression \(1 - \cos^2(θ)\) with the Pythagorean identity. If you subtract \(\cos^2(θ)\) from both sides of the identity, it would give you: \(1 - \cos^2(θ) = \sin^2(θ)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are essential in mathematics as they help to simplify complex trigonometric expressions. One of the most fundamental identities is the Pythagorean identity:

• \(\sin^2(θ) + \cos^2(θ) = 1\)

This identity forms the basis for understanding various other relationships in trigonometry. Knowing how to apply these identities can simplify complicated expressions and solve problems more quickly. Regular practice and familiarity with these identities are key to mastering trigonometry.

Simplify Trigonometric Expressions

Simplifying trigonometric expressions can make complex mathematical problems easier to understand and solve. When given a trigonometric expression such as \(1 - \cos^2(θ)\), our goal is to express it in its simplest form using known identities.The Pythagorean identity tells us that \(\sin^2(θ) + \cos^2(θ) = 1\). By rearranging this, we see that

• \(1 - \cos^2(θ) = \sin^2(θ)\) is an equivalent form.

This illustrates a case where recognizing and applying a basic identity leads directly to a simpler expression. By transforming the original expression into \(\sin^2(θ)\), it often becomes much easier to handle in mathematical calculations and further trigonometric manipulations.

Sine and Cosine Relationship

The relationship between sine and cosine is a core aspect of trigonometry. These two functions are related not only through the Pythagorean identity

• \(\sin^2(θ) + \cos^2(θ) = 1\)

but also through their geometric interpretation in the unit circle. In the unit circle, sine represents the y-coordinate of a point on the circle, while cosine represents the x-coordinate.Understanding how sine and cosine relate to one another through various identities allows for greater flexibility in solving trigonometric problems. For example, the expression \(1 - \cos^2(θ)\) can be immediately transformed into \(\sin^2(θ)\) as per the identity discussed, showcasing the profound connection between these two functions in simplifying expressions.