Problem 13 - Calculus

Question

From the formula for \(\cos (2 t+t)\) find \(\cos 3 t\) in temss of \(\cos t\).

Step-by-Step Solution

Answer

\(\cos 3t = 4\cos^3 t - 3\cos t\). 🌟

Step 1: Apply Angle Sum Formula

We start with the trigonometric identity for the cosine of a sum, \[\cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b\]Using this, we can write:\[\cos(2t + t) = \cos 2t \cdot \cos t - \sin 2t \cdot \sin t\]

Step 2: Use Double Angle Formulas

Now, we apply the double angle identities for cosine and sine in the expression we obtained:\[\cos 2t = 2 \cos^2 t - 1 \quad \text{and} \quad \sin 2t = 2 \sin t \cdot \cos t\]Substituting these into the formula:\[\cos(2t + t) = (2 \cos^2 t - 1) \cdot \cos t - (2 \sin t \cdot \cos t) \cdot \sin t\]

Step 3: Simplify the Expression

Now, let's simplify the expression from the previous step:\[\cos(2t + t) = 2 \cos^3 t - \cos t - 2 \sin^2 t \cdot \cos t\]Using the Pythagorean identity, \(\sin^2 t = 1 - \cos^2 t\), substitute for \(\sin^2 t\):\[\cos(2t + t) = 2 \cos^3 t - \cos t - 2(1 - \cos^2 t) \cdot \cos t\]

Step 4: Final Simplification

Now, simplify further:\[\cos 3t = 2 \cos^3 t - \cos t - 2\cos t + 2 \cos^3 t\]Combine like terms:\[\cos 3t = 4 \cos^3 t - 3 \cos t\]

Step 5: Final Formula

The simplification gives us the formula for \(\cos 3t\) in terms of \(\cos t\):\[\cos 3t = 4 \cos^3 t - 3 \cos t\]

Key Concepts

Angle Sum Formula

The Angle Sum Formula is a key identity in trigonometry. It helps us find the trigonometric values of expression involving sums or differences of angles. The formula for cosine addition is:

• \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \)

This formula breaks down a more complex angle into smaller parts. For example, in the problem of finding \( \cos(3t) \), we rewrite it as \( \cos(2t + t) \). This allows us to use values or identities we know for simpler angles like \( 2t \) and \( t \).
The Angle Sum Formula is powerful because it lets us express things in terms of known quantities. This method can transform a problem into a more manageable form, making complex calculations much easier to handle.

Double Angle Identities

Double angle identities are special trigonometric identities. They specifically relate functions of double angles (like \( 2t \)) to single angles (like \( t \)). The double angle formulas for cosine and sine are as follows:

• For cosine: \( \cos 2t = 2 \cos^2 t - 1 \)
• For sine: \( \sin 2t = 2 \sin t \cdot \cos t \)

These identities simplify tasks like finding \( \cos 3t \), by breaking it down using the angle sum formula \( \cos(2t + t) \). We replace \( \cos 2t \) and \( \sin 2t \) using the double angle identities. This substitution reduces calculations and makes the problem more direct.
Understanding and applying double angle identities can transform expressions and make solving trigonometric equations more straightforward.

Pythagorean Identity

Among the foundational identities in trigonometry, the Pythagorean Identity is crucial. It's written as:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity ties sine and cosine together, showing how they balance on the unit circle. For instance, given \( \sin^2 t \) is needed, we can substitute it with \( 1 - \cos^2 t \) using the identity. In our example, this substitution is used during the simplification process.
The Pythagorean Identity is handy. It helps in transforming equations from sine to cosine form or vice versa. In tasks like our problem, it enables further simplification, making way for a neat and concise solution.

Simplification in Trigonometry

Simplifying trigonometric expressions is a crucial skill. It allows for clearer and more workable forms of complex formulas. In the problem of finding \( \cos 3t \), simplification plays a significant role in reaching the final formula.
The simplification process means combining like terms, substituting identities, and rewriting expressions in simpler forms. In our example, after applying the angle sum and double angle identities, we further reduce the expression through combining terms and using the Pythagorean Identity.

• Combine: \( 2 \cos^3 t - \cos t - 2 \cos t \)
• Simplify: \( 4 \cos^3 t - 3 \cos t \)

This final step provides a simplified, usable formula \( \cos 3t = 4 \cos^3 t - 3 \cos t \). The art of simplification ensures that complex trigonometric expressions are manageable and understandable.