Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{3}=1 $$

Step-by-Step Solution

Answer

The identity is verified because \( (\sin^2 \theta + \cos^2 \theta)^3 = 1 \).

Step 1: Recall the Pythagorean Identity

The Pythagorean Identity states that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is fundamental in trigonometry and is derived from the Pythagorean Theorem.

Step 2: Application of the Identity

Knowing the Pythagorean Identity, we substitute \( 1 \) for \( \sin^2 \theta + \cos^2 \theta \). This gives us \( \left(1\right)^3 = 1 \).

Step 3: Simplify the Expression

To simplify, calculate \( 1^3 \), which equals \( 1 \). Hence, substituting the expression does not change its value. We then have \( 1 = 1 \), verifying the identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the cornerstones of trigonometry. It states that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is directly derived from the Pythagorean Theorem, which relates the sides of a right triangle.
The identity applies to all angles and forms the basis for many trigonometric proofs and simplifications.

• This equation illustrates the constant relationship between the square of the sine and cosine of the same angle.
• It is widely used to simplify expressions and solve trigonometric equations.
• The identity tells us that even though \( \sin \theta \) and \( \cos \theta \) can vary, their squares always sum to 1.

Understanding the Pythagorean Identity helps in tackling various trigonometric problems with ease.

sin^2 theta + cos^2 theta = 1

The equation \( \sin^2 \theta + \cos^2 \theta = 1 \) is an essential part of trigonometry and is commonly referred to as the Pythagorean Identity. It holds true for any angle \( \theta \) because it signifies the inherent property of right-angled triangles.
When you imagine a right triangle inscribed in a unit circle (a circle with radius 1), the sides relate as follows:

• The opposite side is equivalent to \( \sin \theta \).
• The adjacent side corresponds to \( \cos \theta \).

So, by the Pythagorean Theorem, the sum of the squares of these sides (\( \sin^2 \theta + \cos^2 \theta \)) equals the square of the hypotenuse, which in the unit circle is 1.
This identity helps to solve complex trigonometric problems by allowing substitutions and facilitating simplifications. Any trigonometric problem requiring substitution can benefit greatly from understanding this identity.

Trigonometry Fundamentals

Trigonometry is the study of relationships in mathematics involving angles and lengths in triangles. It focuses predominantly on right triangles and the unit circle.
The basics involve understanding three primary functions:

• \( \sin \theta \) (sine) - relates the opposite side of a triangle to its hypotenuse.
• \( \cos \theta \) (cosine) - compares the adjacent side to the hypotenuse.
• \( \tan \theta \) (tangent) - the ratio of the opposite side to the adjacent side.

All these functions are derived from properties of right triangles and the unit circle. These functions form the basis of many complex trigonometric concepts.
The Pythagorean Identity is interwoven into the fundamentals as it provides insight into the proportions of the functions (\( \sin \theta \) and \( \cos \theta \)) for any angle in a triangle. Mastery of these fundamentals enables students to progress to more advanced aspects of trigonometry with confidence.