Problem 27 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{3}=1$$

Step-by-Step Solution

Answer

The identity is verified as true.

Step 1: Recall the Pythagorean Identity

The basic trigonometric identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity holds true for all angles \( \theta \).

Step 2: Apply the Pythagorean Identity

Replace \( \sin^2 \theta + \cos^2 \theta \) with 1, as given by the Pythagorean Identity. This gives us \( (1)^3 \).

Step 3: Simplify the Expression

Calculate \( (1)^3 \). Any number raised to the power of 3 is the number multiplied by itself three times. Hence, \( 1^3 = 1 \).

Step 4: Conclusion

Since the expression simplifies to 1, the left side is equal to the right side of the equation. Therefore, the identity \( (\sin^2 \theta+\cos^2 \theta)^3 = 1 \) is verified.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental concepts in trigonometry. It is given by the equation \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is a reflection of the Pythagorean theorem and holds true for any angle \( \theta \). Its importance cannot be overstated, as it forms the basis for simplifying many trigonometric expressions. By understanding this identity, you're equipped to tackle a wide range of trigonometric problems.

• The identity is derived from the representation of sine and cosine as coordinates of a unit circle.
• In a right triangle, it implies that the sum of the squares of sine and cosine components equals one.
• It allows for the substitution of \( \sin^2 \theta + \cos^2 \theta \) with 1, simplifying expressions significantly.

Next time you come across a complex trigonometric expression, remember the Pythagorean Identity as your tool for simplification.

Simplifying Expressions

Simplifying expressions, especially in trigonometry, often involves substituting complex parts with simpler equivalents. In the context of our problem, once we recognize that \( \sin^2 \theta + \cos^2 \theta = 1 \), simplifying the given expression \((\sin^2 \theta + \cos^2 \theta)^3\) becomes straightforward.

• First, apply known identities like the Pythagorean Identity to reduce complexity.
• Once the identity is applied, which in our case replaces \( \sin^2 \theta + \cos^2 \theta \) with 1, calculating the expression becomes easier.
• The expression \( (1)^3 \) simplifies to 1, showing how powerful identities can be in reducing workload.

Through practice, you'll learn to identify opportunities to simplify, saving time and reducing errors.

Precalculus Concepts

Precalculus serves as the bridge to understanding advanced mathematics. It introduces functions, equations, and identities which are foundational for calculus and beyond. Trigonometric identities like the Pythagorean Identity are crucial in precalculus, helping students to manipulate and solve complex problems more easily.

• Precalculus sets the stage for calculus by solidifying your understanding of functions, angles, and their properties.
• Grasping these concepts empowers you to understand the behavior of graphs and solve equations beyond mere memorization.
• It encourages logical thinking and problem-solving skills by demonstrating how changes in one variable affect another.

Therefore, mastering precalculus concepts isn't just about preparation for calculus, but also for a deep understanding of how mathematical systems interconnect.