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Chapter 6: Problem 3

Use the product-to-sum identities to rewrite each expression. $$\sin 16^{\circ} \cos 20^{\circ}$$

### Short Answer

Expert verified

\[ \sin 16^{\circ} \cos 20^{\circ} = \frac{1}{2} [\sin 36^{\circ} - \sin 4^{\circ}] \]

## Step by step solution

01

## Identify the product-to-sum identity

The given expression is \( \sin 16^{\circ} \cos 20^{\circ}\). To rewrite this using the product-to-sum identities, recall that for any angles \(A\) and \(B\), the identity is: \[ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] \]

02

## Substitute the angles into the identity

Here, \(A = 16^{\circ}\) and \(B = 20^{\circ}\). Substitute these values into the identity: \[ \sin 16^{\circ} \cos 20^{\circ} = \frac{1}{2} [\sin(16^{\circ} + 20^{\circ}) + \sin(16^{\circ} - 20^{\circ})] \]

03

## Simplify the expressions inside the sine functions

Calculate the sums and differences inside the sine functions: \(16^{\circ} + 20^{\circ} = 36^{\circ}\) and \(16^{\circ} - 20^{\circ} = -4^{\circ}\). Thus, the expression becomes: \[ \sin 16^{\circ} \cos 20^{\circ} = \frac{1}{2} [\sin 36^{\circ} + \sin (-4^{\circ})] \]

04

## Use the sine function property for negative angles

The sine of a negative angle can be expressed as \(\sin(-x) = -\sin(x)\). Applying this property, \(\sin(-4^{\circ}) = -\sin(4^{\circ})\). Now, the expression is: \[ \sin 16^{\circ} \cos 20^{\circ} = \frac{1}{2} [\sin 36^{\circ} - \sin 4^{\circ}] \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the angles of triangles to the lengths of their sides. These identities are crucial for simplifying trigonometric expressions and solving trigonometric equations. In the context of this exercise, we are using the product-to-sum identities.

The product-to-sum identities help us transform the product of sine and cosine functions into a sum or difference of sine functions. This transformation simplifies complex expressions and makes them easier to work with.

The specific identity utilized here is: \[ \text{For any angles} \ A \ \text{ and} \ B \,: \ \text{the identity is} \ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] \] This identity allows us to break down the product of sine and cosine into simpler components. Knowing and applying trigonometric identities is essential for anyone studying math, physics, and engineering, as these fields often require the manipulation of trigonometric expressions.

###### Sine and Cosine Functions

Sine and cosine are foundational trigonometric functions that describe the relationship between the angles and sides of right-angled triangles. The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse, whereas the cosine is the ratio of the adjacent side to the hypotenuse.

These functions are periodic and oscillate between \[-1\] and \[1\], with specific values at key angles such as \[0^{\text{\circ}}, 30^{\text{\circ}}, 45^{\text{\circ}}, 60^{\text{\circ}},\] and \[90^{\text{\circ}}\]. Understanding the properties of \[\text{sine}\] and \[\text{cosine}\] , including their graphs and behavior, is crucial to solving trigonometric problems effectively.In the given problem, \[\text{sin} 16^{\text{\circ}} \text{ and cos} 20^{\text{\circ}} \] are multiplied together. By applying the product-to-sum identity, this product is transformed into a sum of simpler sine values, making it easier to handle in subsequent calculations.

Recognizing where and how to apply these identities can significantly streamline the problem-solving process, simplifying complex trigonometric relationships.

###### Angle Addition and Subtraction

Angle addition and subtraction are critical concepts when working with trigonometric functions. These principles allow us to express the sine and cosine of compound angles in terms of the sine and cosine of simple angles. For example, the trigonometric identity for the sine of the sum of two angles is: \[ \text{sin}(A + B) = \text{sin}A \text{cos}B + \text{cos}A \text{sin}B \] Similarly, for the difference of two angles, we have: \[ \text{sin}(A - B) = \text{sin}A \text{cos}B - \text{cos}A \text{sin}B \]These identities help decompose complex angles into simpler components that are easier to evaluate.In the exercise, we use the product-to-sum identity, which incorporates the angle addition and subtraction concepts to rewrite \[\text{sin} 16^{\text{\circ}} \text{cos} 20^{\text{\circ}}\] as a sum of two sine functions. We achieve this by calculating \[ 16^{\text{\circ}} + 20^{\text{\circ}} = 36^{\text{\circ}}\] and \[ 16^{\text{\circ}} - 20^{\text{\circ}} = -4^{\text{\circ}}\]. By recognizing that \[\text{sin}(-4^\text{\circ}) = -\text{sin}(4^\text{\circ})\], the expression simplifies further to \[\frac{1}{2} [\text{sin} 36^\text{\circ} - \text{sin} 4^\text{\circ}]\], illustrating the elegance and utility of these trigonometric principles.

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