Problem 47 - Precalculus: Functions and Graphs

Question

Express \(\sin (u+v+w)\) in terms of trigonometric functions of \(u, v,\) and \(w .\) (Hint: Write $$ \sin (u+v+w) \text { as } \sin [(u+v)+w] $$ and use addition formulas.)

Step-by-Step Solution

Answer

\(\sin(u+v+w) = \sin u \cos v \cos w + \cos u \sin v \cos w + \cos u \cos v \sin w - \sin u \sin v \sin w \).

Step 1: Rewrite the Expression

Start by rewriting the expression for \( \sin(u+v+w) \) as \( \sin((u+v)+w) \). This allows us to apply the addition formula for sine, which is useful for breaking down complex trigonometric expressions.

Step 2: Apply the Sine Addition Formula

Use the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Applying this to \( \sin((u+v)+w) \), we get: \( \sin(u+v) \cos w + \cos(u+v) \sin w \).

Step 3: Break Down \(\sin(u+v)\) and \(\cos(u+v)\)

Further break down \( \sin(u+v) \) and \( \cos(u+v) \) using the sine and cosine addition formulas:- \( \sin(u+v) = \sin u \cos v + \cos u \sin v \)- \( \cos(u+v) = \cos u \cos v - \sin u \sin v \)

Step 4: Substitute Back into the Expression

Substitute the expansions from Step 3 into the expression from Step 2:\((\sin u \cos v + \cos u \sin v) \cos w + (\cos u \cos v - \sin u \sin v) \sin w\)

Step 5: Simplify the Expression

Distribute \( \cos w \) across \( \sin u \cos v + \cos u \sin v \) and \( \sin w \) across \( \cos u \cos v - \sin u \sin v \) to get:\( \sin u \cos v \cos w + \cos u \sin v \cos w + \cos u \cos v \sin w - \sin u \sin v \sin w \)

Key Concepts

Addition Formulas

Addition formulas are mathematical tools used to describe the addition of angles in trigonometry. They play a vital role in simplifying expressions involving sums of angles. These formulas help in breaking down complex trigonometric expressions into more manageable parts. This way, we can handle each part using basic trigonometric values, leading to a simplified solution.

In general, the addition formulas for sine and cosine are part of this toolset. With these formulas, one can evaluate trigonometric functions of summed angles

• The sine addition formula is used to find the sine of a sum of two angles.
• The cosine addition formula does the same but for cosine.

These are particularly useful in physics and engineering, where angles are often used in equations to describe phenomena.

Sine Addition Formula

The sine addition formula is essential for many trigonometric problems. It allows computation of sine for the sum of two angles. The formula states that for two angles, A and B:

• \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

This formula breaks down the sine of a sum into products of sines and cosines of the individual angles.

In the given problem, this formula helped to express \( \sin(u+v+w) \) as \( \sin((u+v)+w) \), by first calculating \( \sin(u+v) \) using the sine addition formula and then applying the formula again to include the third angle, w. This stepwise breakdown enables us to manage more complicated angle expressions efficiently.

Cosine Addition Formula

The cosine addition formula works similarly to the sine addition formula but uses cosines. For two angles, A and B, the formula is expressed as:

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

This formula is useful in finding the cosine of the sum of two angles by breaking it down into manageable parts.

In our solution for \( \sin(u+v+w) \), the cosine addition formula was used to simplify \( \cos(u+v) \) into \( \cos u \cos v - \sin u \sin v \). By doing so, it allows the use of known values or simpler calculations for each of these components, making the overall expression much easier to simplify and solve.

Using both the sine and cosine addition formulas together can unravel even the most complicated trigonometric expressions.