Problem 33 - Precalculus Essentials

Question

In Exercises \(23-34\), verify each identity. $$\sin 4 t=4 \sin t \cos ^{3} t-4 \sin ^{3} t \cos t$$

Step-by-Step Solution

Answer

After applying various trigonometric identities and formulas including double and triple angle formulas, it is confirmed that \(\sin 4t = 4\sin t\cos^3t - 4\sin^3t\cos t\) is a true identity.

Step 1: Use Double Angle Formula

First, recognize that we can express \(\sin 4t\) as \(\sin 2(2t)\). Then apply the double angle formula on the left hand side of the equation. The double angle formula for sine is \(\sin 2A = 2\sin A\cos A\). This will give us \(2\sin 2t\cos 2t\).

Step 2: Apply Double Angle Formula Again

Applying the double angle formula again to \(\sin 2t\) will give us \(2(2\sin t\cos t)\cos 2t\), which simplifies to \(4\sin t\cos t\cos 2t\).

Step 3: Substitute Double angle formula for cosine

One double angle formula for cosine is \(\cos 2t = 1-2\sin^2t\). Apply this formula to the left hand side to help convert it into a combination of sine and cosine. This will give us \(4\sin t\cos t(1 - 2\sin^2t)\).

Step 4: Expand and Simplify

After the successful substitution, let's expand the results. The expression will lead us to \(4\sin t\cos t - 8\sin^3 t\cos t\)

Step 5: Substitute Addition formula for sine

By applying the addition formula on the last term we get \(4\sin t\cos t - 8\sin^3 t\cos t\). Now it looks like the right hand side of the original equation.

Step 6: Rearrange the equation

Rearrange the equated expression to match the given expression. It results in \(4\sin t\cos^3t - 4\sin^3t\cos t\).

Key Concepts

Double Angle Formula

The double angle formula is a powerful tool in trigonometry used to simplify expressions and solve equations. It relates the trigonometric functions of double angles, such as \(2A\), to single angles \(A\). Here, you mainly encounter the formulas for sine and cosine, which are:

• \(\sin 2A = 2\sin A\cos A\)
• \(\cos 2A = \cos^2 A - \sin^2 A\)

These formulas are used to manipulate expressions involving double angles, breaking them down into simpler components. This breakdown makes them easier to work with and often leads to expressions that are more readily compared or simplified. For instance, in solving \(\sin 4t\), we start by using \(\sin 2A\) to break it down step-by-step until it matches the original equation. Whenever you see an angle that is a multiple of another, look to these identities for guidance.

Addition Formula

The addition formulas for sine and cosine are essential in understanding trigonometric expressions involving sums of angles. These formulas help in expanding expressions where angles are added together:

• \(\sin (A + B) = \sin A \cos B + \cos A \sin B\)
• \(\cos (A + B) = \cos A \cos B - \sin A \sin B\)

In the original solution involving \(\sin 4t\), the addition formula aids in validating that the expression is indeed correctly reformulated. By thinking about combining different trigonometric properties, you create a coherent picture where each part of the expression can be understood and verified individually.

Sine and Cosine Product Identities

Sine and cosine product identities allow you to express the product of sine and cosine functions as sums or differences of functions. This method simplifies trigonometric expressions that would otherwise be tough to handle.

• For example, the identity for \(\sin A \cos B\) can be rewritten using sum identities or double angle formulas.

Utilizing these identities, we can address compounded trigonometric parts effectively. In the case of verifying the identity \(\sin 4t = 4 \sin t \cos^3 t - 4 \sin^3 t \cos t\), the ability to manipulate products of sine and cosine is pivotal. This technique further involves substituting and expanding products to fit the structure on the equation's other side. Overall, learning to navigate these identities extends your toolkit for handling complex trigonometry effortlessly.