Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as a sum or difference of trigonometric functions or values. $$2 \sin 58^{\circ} \cos 102^{\circ}$$

Step-by-Step Solution

Answer

\( \sin{160^{\circ}} - \sin{44^{\circ}} \)

Step 1: Identify the Trigonometric Identity

The given expression is of the form \(a \sin{x} \cos{y}\). To simplify it, we'll use the product-to-sum identity: \(\sin{A} \cos{B} = \frac{1}{2} \left[ \sin{(A+B)} + \sin{(A-B)} \right]\). However, here we have \(2 \sin{58^{\circ}} \cos{102^{\circ}}\), so we should adjust the identity to: \(2 \sin{A} \cos{B} = \sin{(A+B)} + \sin{(A-B)}\).

Step 2: Apply Trigonometric Identity

By substituting \(A = 58^{\circ}\) and \(B = 102^{\circ}\) into the identity \(2 \sin{A} \cos{B} = \sin{(A+B)} + \sin{(A-B)}\), we get:\[ 2 \sin{58^{\circ}} \cos{102^{\circ}} = \sin{(58^{\circ} + 102^{\circ})} + \sin{(58^{\circ} - 102^{\circ})} \]

Step 3: Simplify the Expression

Calculate \(58^{\circ} + 102^{\circ} = 160^{\circ}\) and \(58^{\circ} - 102^{\circ} = -44^{\circ}\). Substitute these into the equation:\[ \sin{160^{\circ}} + \sin{(-44^{\circ})} \]

Step 4: Understand Negative Angles in Sine Function

Recall that \( \sin{(-\theta)} = -\sin{\theta} \). Thus, \( \sin{(-44^{\circ})} = -\sin{44^{\circ}} \). Substitute back into the expression to get:\[ \sin{160^{\circ}} - \sin{44^{\circ}} \]

Step 5: Final Expression

Therefore, the original expression \(2 \sin{58^{\circ}} \cos{102^{\circ}}\) can be written as the sum: \[ \sin{160^{\circ}} - \sin{44^{\circ}} \].

Key Concepts

Product-to-Sum Identities

In trigonometry, converting products of sine and cosine into sums can often simplify calculations. This transformation is achieved using the product-to-sum identities, which help break down a complex multiplication into simpler addition or subtraction expressions.
\[\sin{A} \cos{B} = \frac{1}{2} [\sin(A + B) + \sin(A - B)]\]
In the exercise provided, we had to handle an expression of the form \(2 \sin{A} \cos{B}\). By tweaking the identity, we simplify it as follows:
- Multiply both sides of the identity by 2.
Now it becomes:
\[2 \sin{A} \cos{B} = \sin(A + B) + \sin(A - B)\]
This adjustment allows us to work directly with the given terms without having to divide by a constant later. Using these identities helps decomposing seemingly daunting trigonometric products into more manageable sums or differences. This is particularly useful in calculus or solving trigonometric equations.

Negative Angle in Trigonometry

Understanding how negative angles work in trigonometry is essential because angles can be measured in both positive (counter-clockwise) and negative (clockwise) directions. This understanding influences how trigonometric functions behave.
Specifically for sine, one of the key properties is:

• \( \sin(-\theta) = -\sin(\theta) \)

This tells us that sine is an odd function, meaning it becomes the negative counterpart of its positive angle counterpart.
In the solution provided, after calculating \(58^{\circ} - 102^{\circ} = -44^{\circ}\), we applied this rule. This transformation led to:

• \(\sin(-44^{\circ}) = -\sin(44^{\circ})\)

By understanding these properties, we can simplify expressions efficiently, altering negative angles into conventional positive forms which are often easier to recognize and work with.

Simplification Techniques

When faced with trigonometric expressions, simplification can make an otherwise complex problem easier to manage. Using identities, handling different angle signs, and performing precise arithmetic manipulations are crucial techniques in trigonometry.
To simplify:\First, ensure every part of the expression adheres to standard forms. Employ identities like product-to-sum as in the case here, allowing a transition from multiplication to a combination of simpler sums.
Second, handle negative angles appropriately; negative sine translations turn them into a more recognizable pattern. Recognize and utilize characteristics like:\- \(\sin(-\theta) = -\sin(\theta)\)
Lastly, make sure to perform calculations meticulously to achieve readability in your final solution. Calculations as simple as \(58^{\circ} + 102^{\circ} = 160^{\circ}\) illustrate how clear and straightforward arithmetic culminates in complete simplification, yielding a neat expression such as \(\sin{160^{\circ}} - \sin{44^{\circ}}\).
Simplification not only improves understanding but also aids in applying trigonometry concepts across different math areas.