Question

Write the following equation in the form \(y=A \sin B x+D,\) where \(A, B,\) and \(D\) are constants: \(y=6 \sin x \cos ^{3} x+6 \sin ^{3} x \cos x-3\).

Short answer

y = 3 \sin 2x - 3

Step-by-step solution

- Identify Trigonometric Identities

Recognize that we can use trigonometric identities to simplify the equation. The expressions \(\sin x \cos^{3} x\) and \(\sin^{3} x \cos x\) can be simplified using the product-to-sum identities.

- Simplify \(\sin x \cos^{3} x\)

Rewrite \(\sin x \cos^{3} x\) using the identity \(\cos^{3} x = (\cos x)(\cos^{2} x) = (\cos x)(1 - \sin^{2} x)\). This gives us: \[\sin x \cos^{3} x = \sin x \(\cos x - \cos x \sin^{2} x\) = \sin x \cos x - \sin^{3} x \cos x\].

- Combine Similar Terms

Combine the terms involving \(\sin x \cos x\) from the simplified expression in Step 2 with \(6 \sin^{3} x \cos x\). This results in: \[6 \sin x \cos^{3} x + 6 \sin^{3} x \cos x = 6 (\sin x \cos x - \sin^{3} x \cos x) + 6 \sin^{3} x \cos x = 6 \sin x \cos x\].

- Insert the Simplified Term Back into the Equation

Replace the simplified term back into the original equation, resulting in: \[y = 6 \sin x \cos x - 3\].

- Use Product-to-Sum Identity

Apply the product-to-sum identity to the term \(6 \sin x \cos x\). According to the identity: \(\sin x \cos x = \frac{1}{2} \sin 2x\), we have: \[- y = 6 \frac{1}{2} \sin 2x - 3 = 3 \sin 2x - 3 -\].

- Rewrite into Desired Form

Now, express the final result in the desired form \(y = A \sin B x + D\). Thus we can clearly see that \[A = 3, \ B = 2, \ D = -3. -\]

Key concepts

Product-to-Sum Identity

In trigonometry, product-to-sum identities are useful tools for simplifying products of sine and cosine functions. These identities convert products into sums or differences of trigonometric functions, making them easier to handle.
For instance, the identity for \(\sin x \cos x\) is:
\[ \sin x \cos x = \frac{1}{2} \sin 2x \]
Here's why this is useful:
When faced with a term like \(6 \sin x \cos x\), applying the product-to-sum identity simplifies it to \(3 \sin 2x\).
This reduces the complexity of the expression, making further calculations more straightforward.
The key takeaway is to recognize patterns where these identities can be applied for simplification.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex terms into simpler ones through the use of various identities. Here are some techniques commonly used:

Identify Common Trigonometric Identities:
Recognizing patterns such as Pythagorean identities or angle sum and difference identities helps in simplifying expressions.

Rewrite Using Fundamental Identities:
For example, rewrite \(\cos^2 x\) as \(1 - \sin^2 x\) when simplifying expressions.

Combine Similar Terms:
Once simplified, terms with common factors can often be combined. For instance, combining \(\sin x \cos x - \sin^3 x \cos x + \sin^3 x \cos x\) results in \(\sin x \cos x\).

Simplifying trigonometric expressions makes solving complex equations easier and often reveals a more straightforward path to the solution.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions like sine, cosine, and tangent. Solving these equations often requires a deep understanding of trigonometric identities and algebraic manipulation.
To solve a trigonometric equation such as \(y=6 \sin x \cos^3 x + 6 \sin^3 x \cos x - 3\), follow these steps:

• Simplify Each Term: Use identities like the product-to-sum identities to simplify trigonometric products.
• Combine Like Terms: Group similar terms to reduce the equation to a simpler form.
• Apply Identities: Use known identities to transform the terms into a desired format, like \(A \sin B x + D\).

Using these methods, we simplified the original equation to \(y = 3 \sin 2x - 3\), identifying constants: \(A = 3, B = 2, D = -3\).
Understanding these techniques is crucial for mastering trigonometric equations.