Problem 56 - Precalculus: Functions and Graphs

Question

Exer. \(55-56:\) (a) Compare the decimal approximations of both sides of equation ( 1). (b) Find the acute angle \(x\) such that equation ( 2) is an identity. (c) How does equation ( 1) relate to equation ( 2)\(?\) (1) \(\sin 35^{\circ}+\sin 25^{\circ}=\cos 5^{\circ}\) \((2) \sin (\alpha+\beta)+\sin (\alpha-\beta)=\cos \beta\)

Step-by-Step Solution

Answer

(a) Both sides equal approximately 0.9962. (b) Angle \(x\) is \(30^{\circ}\). (c) Eq. (1) is a specific case of Eq. (2).

Step 1: Understanding Problem (a)

In this step, we need to compare the decimal approximations of both sides of equation (1), which is \( \sin 35^{\circ} + \sin 25^{\circ} = \cos 5^{\circ} \). We'll calculate the decimal values of each side.

Step 2: Calculate Decimal Approximations

Calculate \( \sin 35^{\circ} \approx 0.5736 \), \( \sin 25^{\circ} \approx 0.4226 \), and \( \cos 5^{\circ} \approx 0.9962 \). Add the first two to get \( 0.5736 + 0.4226 = 0.9962 \). Compare both sides: Left side \( = 0.9962 \) and Right side \( = 0.9962 \). They are equal.

Step 3: Understanding Problem (b)

For this problem, we need to determine the acute angle \(x\) such that equation (2) becomes an identity. Equation (2) is \( \sin (\alpha+\beta) + \sin (\alpha-\beta) = \cos \beta \).

Step 4: Relating Equation 1 to 2

Note that equations (1) and (2) are similar if we set \( \alpha = 30^{\circ} \) and \( \beta = 5^{\circ} \). Thus, the angle \( x \) is close to these values.

Step 5: Finding Acute Angle x

In Equation (2), specifically solve for which \(x\), if set \( \alpha = 30^{\circ} \) and \( \beta = 5^{\circ} \), the identity holds. Here, the setup corresponds to equality like in equation (1). Thus, \( x = 30^{\circ} \).

Step 6: Understanding Problem (c)

Equation (1) is a specific numerical example of the more general form presented in equation (2), with specific values substituted for \( \alpha \) and \( \beta \). Essentially, equation (1) exemplifies equation (2) when \( \alpha = 30^{\circ} \) and \( \beta = 5^{\circ} \).

Key Concepts

Decimal Approximations

Decimal approximations are crucial in comparing equations involving trigonometric identities. Essentially, this involves finding the approximate decimal values of trigonometric functions, like sine and cosine. For instance, in equation (1), we have \( \sin 35^{\circ} + \sin 25^{\circ} = \cos 5^{\circ} \). To check if both sides are equal, we convert these trigonometric ratios to decimals.

For \( \sin 35^{\circ} \), the value is approximately \( 0.5736 \) and \( \sin 25^{\circ} \) is nearly \( 0.4226 \). Adding these gives \( 0.5736 + 0.4226 = 0.9962 \). On the right-hand side, \( \cos 5^{\circ} \) evaluates to \( 0.9962 \).

By getting these decimal approximations, students can see that both sides of the equation balance out to the same value: \( 0.9962 \). This shows that the equation is correct, providing a straightforward way to verify complex trigonometric identities.

Acute Angle

In trigonometry, understanding angles is crucial, especially when dealing with equations. An acute angle is an angle less than \( 90^{\circ} \). In exercise 56-b, we are tasked with discovering the acute angle \( x \) that solves equation (2).

This equation \( \sin (\alpha+\beta) + \sin (\alpha-\beta) = \cos \beta \) becomes an identity when particular values satisfy it.

• An identity means that the equation holds true for all values of the variable involved, demonstrating consistent truth across the board.
• By setting \( \alpha = 30^{\circ} \) and \( \beta = 5^{\circ} \), the equation morphs into the format that mirrors equation (1).

When you substitute these into the given identity, you achieve equality similar to that found in part (a). As a result, the acute angle \( x \) that fulfills this identity is \( 30^{\circ} \). Understanding how \( x = 30^{\circ} \) makes equation (2) true underlines the interconnected nature of these trigonometric expressions.

Equation Comparison

Comparing equations is a key skill, especially when dealing with trigonometric identities. In exercises like this, recognizing how equations can relate to one another enhances comprehension.

Equation (1), \( \sin 35^{\circ} + \sin 25^{\circ} = \cos 5^{\circ} \), is a particular instance of the more general equation (2): \( \sin (\alpha+\beta) + \sin (\alpha-\beta) = \cos \beta \).

• To see the connection, set \( \alpha = 30^{\circ} \) and \( \beta = 5^{\circ} \).
• This substitution shows that equation (1) is not so different from equation (2), but rather a specific case with fixed angle values.

Understanding this relation aids students in grasping how individual equations can be derived from more general mathematical statements. It also highlights how certain combinations of angles work together to form identities, offering deeper insights into the symmetry and patterns in trigonometry.