Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-16: Express as a trigonometric function of one angle. $$ \sin 57^{\circ} \cos 4^{\circ}+\cos 57^{\circ} \sin 4^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 61^{\circ} \)

Step 1: Identify the Trigonometric Formula

The given expression \( \sin 57^{\circ} \cos 4^{\circ} + \cos 57^{\circ} \sin 4^{\circ} \) matches the format of the sine addition formula for two angles, which is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). In this problem, \( A = 57^{\circ} \) and \( B = 4^{\circ} \).

Step 2: Apply the Sine Addition Formula

Use the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Replace \( A \) with \( 57^{\circ} \) and \( B \) with \( 4^{\circ} \) in the formula. This gives us \( \sin(57^{\circ} + 4^{\circ}) = \sin 57^{\circ} \cos 4^{\circ} + \cos 57^{\circ} \sin 4^{\circ} \).

Step 3: Simplify

The expression \( \sin(57^{\circ} + 4^{\circ}) \) simplifies to \( \sin(61^{\circ}) \) since \( 57^{\circ} + 4^{\circ} = 61^{\circ} \). Thus, the expression \( \sin 57^{\circ} \cos 4^{\circ} + \cos 57^{\circ} \sin 4^{\circ} \) is equivalent to \( \sin 61^{\circ} \).

Key Concepts

Sine Addition Formula

The sine addition formula is an important identity in trigonometry that allows us to express the sine of the sum of two angles in terms of the sine and cosine of the individual angles. Mathematically, it is expressed as: \[\sin(A + B) = \sin A \cos B + \cos A \sin B\]This formula is extremely helpful for simplifying expressions where angles are being added together.
In the context of our exercise, we recognize that the expression \( \sin 57^{\circ} \cos 4^{\circ} + \cos 57^{\circ} \sin 4^{\circ} \) matches this formula. This means we can substitute \( A = 57^{\circ} \) and \( B = 4^{\circ} \) to directly simplify the given trigonometric expression.

Angle Addition in Trigonometry

Angle addition is a common operation in trigonometry where two angle measures are added together. For trigonometric functions, the addition of angles often leads to the use of addition formulas, such as the sine addition formula.

• These formulas help in transforming and simplifying trigonometric expressions.
• The sum \( A + B \) results in a single angle with its own trigonometric function, making calculations simpler.

By applying angle addition to \( 57^{\circ} + 4^{\circ} \), the problem demonstrates how this operation is not only theoretical but practical in simplifying expressions like the one given. The computed angle \( 61^{\circ} \) is the lynchpin for further simplification using the sine addition identity.

Trigonometric Simplification

Trigonometric simplification is the process of making trigonometric expressions more manageable or easier to evaluate. It leverages identities and formulas to reduce complex expressions into simpler forms.In this specific exercise:

• The original complex expression \( \sin 57^{\circ} \cos 4^{\circ} + \cos 57^{\circ} \sin 4^{\circ} \) is reduced.
• We utilized the sine addition formula to transform it into the much simpler form \( \sin 61^{\circ} \).

This transformation illustrates the power of trigonometric identities in condensing information, facilitating easier computation, and understanding the relationships between angles and their functions.