Problem 21 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\sin ^{4} t=\frac{3}{8}-\frac{1}{2} \cos 2 t+\frac{1}{8} \cos 4 t\)

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \frac{3}{8} - \frac{1}{2} \cos 2t + \frac{1}{8} \cos 4t \).

Step 1: Express \( \sin^4 t \) in Terms of Cosines

We start by expressing \( \sin^2 t \) using the double-angle identity: \( \sin^2 t = \frac{1 - \cos 2t}{2} \). Then, square this expression to obtain \( \sin^4 t \):\[\sin^4 t = \left( \frac{1 - \cos 2t}{2} \right)^2 = \frac{(1 - \cos 2t)^2}{4}.\]

Step 2: Expand the Squared Expression

Expand \((1 - \cos 2t)^2\) to simplify it.\[(1 - \cos 2t)^2 = 1 - 2\cos 2t + \cos^2 2t.\]So,\[\sin^4 t = \frac{1 - 2\cos 2t + \cos^2 2t}{4}.\]

Step 3: Use the Double-Angle Identity for \( \cos^2 2t \)

Use the identity \( \cos^2 A = \frac{1 + \cos 2A}{2} \) with \( A = 2t \) to express \( \cos^2 2t \):\[\cos^2 2t = \frac{1 + \cos 4t}{2}.\]

Step 4: Substitute and Simplify

Substitute \( \cos^2 2t = \frac{1 + \cos 4t}{2} \) back into the equation:\[\sin^4 t = \frac{1 - 2\cos 2t + \frac{1 + \cos 4t}{2}}{4}.\]Simplify the expression:\[\sin^4 t = \frac{1 - 2\cos 2t + \frac{1}{2} + \frac{\cos 4t}{2}}{4} = \frac{3}{8} - \frac{1}{2}\cos 2t + \frac{1}{8}\cos 4t.\]

Step 5: Conclude the Verification

Since we have successfully transformed \( \sin^4 t \) to match the right side of the given equation, we have verified that the identity is indeed correct.

Key Concepts

Understanding Double-Angle Identities

The double-angle identities are powerful tools in trigonometry, useful for simplifying expressions and solving equations. The most common forms are for sine and cosine:

• For sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• Alternatively for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• And: \( \cos 2\theta = 1 - 2\sin^2 \theta \)

These identities allow us to express squares of sine and cosine functions in terms of double angles. This is particularly useful in proving identities. For example, in this exercise, \( \sin^2 t \) is expressed as \( \frac{1 - \cos 2t}{2} \), making it easier to work with \( \sin^4 t \) through squaring the expression and further simplification.

Trigonometric Functions Basics

Trigonometric functions like sine, cosine, and tangent are foundational elements in mathematics, particularly used to relate angles to side lengths in right-angled triangles. These functions also extend to periodic waveforms useful in modeling cyclical phenomena:

• \( \sin \theta \) relates the ratio of the length of the opposite side to the hypotenuse.
• \( \cos \theta \) relates the adjacent side to the hypotenuse.
• Both functions are periodic, with periods decreasing as functions' argument angles increase.

Higher powers of these functions, such as \( \sin^4 t \), can be challenging to work with directly, so using identities to transform these into more manageable forms greatly aids in solving equations and verifying identities.

Verification of Trigonometric Identities

Verification of identities involves proving that two expressions are equivalent. This is a crucial part of trigonometry as it allows mathematicians to build more complex theories on solid foundational truths. To verify a trigonometric identity:

• Simplify one or both sides of the equation, typically using algebraic manipulation and known identities.
• Use strategic substitutions and transformations to create easier-to-work-with forms. For example, our exercise uses square transformations and double-angle identities to simplify the given equation for sine.
• Conclude by showing each step achieves equivalency with the given target equation.

In this exercise, simplification using identities transforms a complicated powers expression into the identical equation as specified, confirming the validity of the identity. Verifying identities not only builds problem-solving proficiency but solidifies understanding of how trigonometric principles work.