Problem 18 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\sin 4 t=4 \sin t \cos t\left(1-2 \sin ^{2} t\right)\)

Step-by-Step Solution

Answer

The identity \( \sin 4t = 4 \sin t \cos t (1 - 2 \sin^2 t) \) is verified.

Step 1: Understand the Trigonometric Identity

We need to determine whether \( \sin 4t \) can be expressed as \( 4 \sin t \cos t (1 - 2 \sin^2 t) \). We recognize that \( \sin 4t \) is double the angle formula for sine applied twice.

Step 2: Use the Double Angle Formula

The double angle formula for sine is \( \sin 2\theta = 2 \sin \theta \cos \theta \). First, rewrite \( \sin 4t \) as \( \sin(2(2t)) \), which gives us \( \sin 4t = 2 \sin 2t \cos 2t \).

Step 3: Apply the Double Angle Formula Iteratively

We know \( \sin 2t = 2 \sin t \cos t \). Substitute this into the expression: \( \sin 4t = 2 (2 \sin t \cos t) \cos 2t = 4 \sin t \cos t \cos 2t \).

Step 4: Express \( \cos 2t \) Using Trigonometric Identities

The double angle formula for cosine is \( \cos 2t = 1 - 2 \sin^2 t \). Replace \( \cos 2t \) in the expression: \( \sin 4t = 4 \sin t \cos t (1 - 2 \sin^2 t) \).

Step 5: Compare Both Sides

Notice that after the substitutions, both expressions match: \( \sin 4t = 4 \sin t \cos t (1 - 2 \sin^2 t) \). Hence the identity is verified.

Key Concepts

Double Angle Formulas

Trigonometric identities are powerful tools that help us simplify or transform trigonometric expressions. Double Angle Formulas are specific types of identities that are useful when the angle in question is double some value. For sine and cosine, the double angle formulas are:

• For sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• An alternate form for cosine: \( 1 - 2 \sin^2 \theta \)

These formulas are derived from the unit circle and the fundamental definitions of sine and cosine. They are very useful for expressing trigonometric functions at larger angles in terms of smaller angles. In the exercise, we applied the double angle formula twice to express \( \sin 4t \). First, we applied it for \( \sin 2t \) and then used it again to solve \( \sin 4t = \sin(2(2t)) \), resulting in \( 2 \sin 2t \cos 2t \).

Sine Function

The sine function is one of the basic trigonometric functions, primarily dealing with the length of the opposite side to a given angle in a right triangle divided by the hypotenuse.
It is periodic with a period of \( 2\pi \), meaning the function repeats every \( 2\pi \).
Some key points about the sine function are:

• Its value ranges from -1 to 1.
• It is an odd function, so \( \sin(-\theta) = -\sin(\theta) \).
• Its graph is a wave that oscillates up and down crossing the origin.

In relation to our exercise, understanding these properties helps in visualizing how repeated applications of specific identities transform the function. This is especially important when using the Double Angle Formula as splitting or doubling the angle impacts the sine value effectively by reshaping the triangle’s geometry. This is why, as shown in the exercise, to express \( \sin 4t \) we begin by doubling \( \sin 2t \).

Cosine Function

The cosine function, another fundamental trigonometric function, usually represents the length of the adjacent side to a given angle in a right triangle divided by the hypotenuse.
It is also periodic, with a period of \( 2\pi \), similar to sine, repeating each revolution:

• Its range is from -1 to 1.
• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• The graph of cosine starts its cycle at 1, peaks, troughs, and comes back to the starting point.

In verifying the given trigonometric identity, we use the cosine's double angle formula to express \( \cos 2t \) as \( 1 - 2 \sin^2 t \). This step is crucial as it allows us to rewrite the expression in terms of sine, ultimately facilitating the verification process by aligning the components to match both sides of the identity. This shows how cosine’s properties are leveraged to manipulate trigonometric equations effectively through identities.