Problem 44 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ \sin ^{2} \theta-\cos ^{2} \theta=2 \sin ^{2} \theta-1 $$

Step-by-Step Solution

Answer

By using the Pythagorean Identity \(\sin^2\theta + \cos^2\theta = 1\), \(\cos^2\theta\) can be written as \(\sin^2\theta\) term. After applying this substitution, the left side of the equation has been transformed into the right side, \(2\sin^{2}\theta - 1\).

Step 1: Use Pythagorean Identity

Use the Pythagorean Identity \(\sin^2\theta + \cos^2\theta = 1\). This identity allows for the transformation of \(\sin^2\theta\) terms into \(\cos^2\theta\) terms or vice versa. Here, it will be helpful to modify \(\cos^2\theta\) in the left side of the given equation into a \(\sin^2\theta\) term which makes the left side look similar to the right side. Therefore, replace \(\cos^2\theta = 1 - \sin^2\theta\) in the equation which results in \(\sin ^{2} \theta-(1 - \sin ^{2} \theta)=2 \sin ^{2} \theta-1\)

Step 2: Simplify the Equation

Now, simplify the equation established in step 1. This involves adding the like terms and simplifying into a more concise equation. The result is \(2\sin^{2}\theta -1 = 2\sin^{2}\theta - 1\)

Step 3: Check the Equation

Finally, always check if the transformed left side is now equal to the right side. Here, it is observed that \(2\sin^{2}\theta - 1 = 2\sin^{2}\theta - 1\), thus the left side has indeed been successfully transformed into the right side.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric identity. It states that for any angle \( \theta \), the sum of the square of sine and cosine is always equal to 1. Mathematically, it is expressed as \( \sin^2\theta + \cos^2\theta = 1 \). This identity is extremely useful in trigonometry for transforming and simplifying equations.
For instance, in our exercise, it helped us replace the \( \cos^2\theta \) term. By using the identity, we can express \( \cos^2\theta \) as \( 1 - \sin^2\theta \). This helps in transforming and simplifying trigonometric expressions, especially when we want all terms to be in one trigonometric function such as sine.
The Pythagorean Identity arises from the Pythagorean theorem, which deals with the relationship of the sides within a right triangle. Similarly, \( \sin\theta \) and \( \cos\theta \) represent the ratios of the sides of a right-angled triangle, leading beautifully to this identity.

trigonometric transformation

Trigonometric transformation involves altering expressions using trigonometric identities and formulas to simplify or solve them. This process is crucial when dealing with equations that need to be transformed into a more manageable or comparable form. In the given exercise, we used a trigonometric transformation to address the left side of the equation and simplify it to match the right side.

• By transforming \( \cos^2\theta \) to \( 1 - \sin^2\theta \) using the Pythagorean Identity, a transformation occurs, making it easier to combine the terms.
• Such transformations are necessary to unify expressions under similar terms or to prepare them for further simplification or solutions.
• Through trigonometric transformations, complex problems often become more approachable and solvable.

These transformations hold significant power in mathematical problem-solving, enabling the reshaping or reworking of trigonometric expressions into desired forms that align with mathematical objectives.

sin^2 and cos^2 conversion

The conversion between \( \sin^2\theta \) and \( \cos^2\theta \) is a key trigonometric technique used to simplify and solve equations. This conversion is greatly facilitated by the Pythagorean Identity. In our example, we utilized this conversion technique effectively.

• Converted \( \cos^2\theta \) to \( 1 - \sin^2\theta \).
• This conversion made it possible to look at equations entirely in terms of \( \sin^2\theta \).
• Similar conversions can be done the other way around, transforming \( \sin^2\theta \) into \( 1 - \cos^2\theta \).

By converting between these two terms, you gain flexibility in solving equations where having a uniform trigonometric function is beneficial.
This technique is particularly useful when you need to match expressions, substitute variables, or verify trigonometric identities. Therefore, understanding how to perform these conversions is crucial in mastering trigonometry and tackling a variety of mathematical challenges.