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Chapter 8: Problem 41

Establish each identity. $$3 \sin ^{2} \theta+4 \cos ^{2} \theta=3+\cos ^{2} \theta$$

Short Answer

Expert verified
The identity is verified as both sides simplify to \( 3 + \cos^2 \theta \).

Step by step solution

01

- Rewrite Using Pythagorean Identity

First, recall the Pythagorean identity:
02

- Replace \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \)

Substitute the identity \( \sin^2 \theta = 1 - \cos^2 \theta \) into the left side of the equation: \[ 3 \sin^2 \theta + 4 \cos^2 \theta = 3 (1 - \cos^2 \theta) + 4 \cos^2 \theta \]
03

- Simplify the Left Side

Distribute the 3 on the left side and combine like terms: \[ 3 - 3\cos^2 \theta + 4\cos^2 \theta = 3 + \cos^2 \theta \]
04

- Confirm the Identity

Observe that \( -3\cos^2 \theta + 4\cos^2 \theta \) simplifies to \( \cos^2 \theta \): \[ 3 + \cos^2 \theta = 3 + \cos^2 \theta \] Thus, the identity is established.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Identity
The Pythagorean Identity is a fundamental relation in trigonometry. It states that for any angle \(\theta\), the following equation holds: \[ \sin^2 \theta + \cos^2 \theta = 1 \].
This identity is derived from the Pythagorean theorem applied to the unit circle.
It provides a way to express either \( \sin^2 \theta \) or \( \cos^2 \theta \) in terms of the other.
In our exercise, this is used to manipulate and prove given trigonometric expressions.
By substituting one variable with its equivalent, we simplify and solve trigonometric equations more easily.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions often involves using basic identities such as the Pythagorean Identity.
This simplification process can make complex problems more manageable.
In the given exercise, to simplify \(3 \sin^2 \theta + 4 \cos^2 \theta\), we use the identity \( \sin^2 \theta = 1 - \cos^2 \theta\).
Substituting \(\sin^2 \theta \) in the expression, the equation becomes: \[ 3(1 - \cos^2 \theta) + 4 \cos^2 \theta \].
This transforms a complex trigonometric expression into a simpler form, which can then be combined and reduced to the desired result.
Trigonometric Substitution
Trigonometric substitution is a powerful technique used for simplifying expressions and proving identities.
In this technique, one trigonometric function is replaced with another using known identities.
In our example, \( \sin^2 \theta \) was substituted with \(1 - \cos^2 \theta\), thanks to the Pythagorean Identity.
This substitution allows us to convert the original equation into something more straightforward:
\[ 3(1 - \cos^2 \theta) + 4 \cos^2 \theta \].
By distributing and combining like terms, the expression simplifies significantly, helping to establish the identity.
Mathematical Proof
Mathematical proof is a logical process used to demonstrate the truth of a statement.
In trigonometry, proving identities often requires step-by-step transformations using known identities and algebraic manipulation.
For our exercise, we started with the given equation, applied the Pythagorean Identity, performed substitution, and simplified the expression.
Through each step, we showed that both sides of the equation are identical:
\[ 3 - 3 \cos^2 \theta + 4 \cos^2 \theta = 3 + \cos^2 \theta \].
This consistent logical process confirms the validity of the identity, demonstrating the importance of clear and systematic proof in mathematics.

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