Problem 32 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$(a \cos t-b \sin t)^{2}+(a \sin t+b \cos t)^{2}=a^{2}+b^{2}$$

Step-by-Step Solution

Answer

The identity is verified; the expressions are equal to \(a^2 + b^2\).

Step 1: Expand the First Term

Let's expand the first term: \((a \cos t - b \sin t)^2\). This expression can be expanded as follows:\[(a \cos t - b \sin t)^2 = (a \cos t)^2 - 2(a \cos t)(b \sin t) + (b \sin t)^2\]Simplifying, we get:\[a^2 \cos^2 t - 2ab \cos t \sin t + b^2 \sin^2 t\]

Step 2: Expand the Second Term

Now, let's expand the second term:\((a \sin t + b \cos t)^2\). This expression can be expanded as follows:\[(a \sin t + b \cos t)^2 = (a \sin t)^2 + 2(a \sin t)(b \cos t) + (b \cos t)^2\]Simplifying, we get:\[a^2 \sin^2 t + 2ab \sin t \cos t + b^2 \cos^2 t\]

Step 3: Combine Both Expanded Terms

Combine both expanded terms from Step 1 and Step 2:\[(a \cos t - b \sin t)^2 + (a \sin t + b \cos t)^2 = a^2 \cos^2 t - 2ab \cos t \sin t + b^2 \sin^2 t + a^2 \sin^2 t + 2ab \sin t \cos t + b^2 \cos^2 t\]Observe that the \(-2ab \cos t \sin t\) and \(+2ab \sin t \cos t\) terms cancel each other out.

Step 4: Use Pythagorean Identity

After cancellation, we are left with:\[a^2 \cos^2 t + b^2 \sin^2 t + a^2 \sin^2 t + b^2 \cos^2 t\]Let's group the terms:\[a^2 (\cos^2 t + \sin^2 t) + b^2 (\sin^2 t + \cos^2 t)\]Using the Pythagorean identity: \(\cos^2 t + \sin^2 t = 1\), this simplifies to:\[a^2(1) + b^2(1) = a^2 + b^2\]

Step 5: Conclusion: Verification of Identity

Thus, we have verified that:\[(a \cos t - b \sin t)^2 + (a \sin t + b \cos t)^2 = a^2 + b^2\]This confirms that the given expression is indeed an identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental trigonometric identities. It stems from the basic relationship in a right triangle, where the sum of the squares of the two legs equals the square of the hypotenuse. In trigonometry, this relationship is expressed as:\[\cos^2 t + \sin^2 t = 1\]This identity enables simplification and transformation of trigonometric expressions, like in our exercise, where we switched:

• \(a^2 (\cos^2 t + \sin^2 t)\) into \(a^2 \times 1\), simplifying it to \(a^2\)
• \(b^2 (\sin^2 t + \cos^2 t)\) into \(b^2 \times 1\), simplifying it to \(b^2\)

Understanding the Pythagorean identity is crucial as it allows conversion of complex trigonometric forms into simpler versions, facilitating easier calculations and verifications of identities.

Expansion of Binomials

The expansion of binomials is a common task in algebra that involves expanding expressions raised to a power. Let's take a binomial of the form \((x + y)^2\). To expand:

• Square the first term: \(x^2\)
• Multiply the two terms together and double the result: \(2xy\)
• Square the second term: \(y^2\)

This gives the expansion:\[x^2 + 2xy + y^2\]In our exercise, we used this method to expand trigonometric terms like \((a \cos t - b \sin t)^2\), applying similar rules to include trigonometric functions. By distributing and simplifying, identities or balanced equations are formed.

Verification of Identities

Verification of trigonometric identities involves proving two expressions are equal. To verify, follow these steps:

• Simplify each side of the identity separately.
• Use known identities, such as the Pythagorean identity, to transform terms.
• Cancel identical terms or combine like terms.
• Finally, check if the simplified expressions are the same on both sides.

In our problem, expanding and simplifying \((a \cos t - b \sin t)^2 + (a \sin t + b \cos t)^2\) followed by combining terms, supported the conclusion that both sides equaled \(a^2 + b^2\). With each step verified, achieving equal results on both sides confirmed the given statement as an identity.

Trigonometric Functions

Trigonometric functions like sine and cosine are foundational in understanding many mathematical concepts. They describe the relationship between the angles and sides of a right triangle, and are periodic functions with specific properties:

• Sine (\(\sin t\)): Represents the y-coordinate of a point on the unit circle.
• Cosine (\(\cos t\)): Represents the x-coordinate of a point on the unit circle.
• These functions cycle through their values as angles increase, which is useful in modeling periodic phenomena.

In our exercise, using trigonometric functions within binomials offered an approach to manipulate and prove the identity. Understanding these functions is key to following the expansion and simplification processes in exercises like verifying identities.