Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin 40^{\circ} \cos 50^{\circ}+\cos 40^{\circ} \sin 50^{\circ}$$

Step-by-Step Solution

Answer

The exact value of the expression is 1.

Step 1: Identify the Trigonometric Identity

Notice that the expression \( \sin 40^{\circ} \cos 50^{\circ} + \cos 40^{\circ} \sin 50^{\circ} \) matches the form of the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Here, we assign \( a = 40^{\circ} \) and \( b = 50^{\circ} \).

Step 2: Apply the Sine Addition Formula

Using the identity \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), replace \( a \) and \( b \) with \( 40^{\circ} \) and \( 50^{\circ} \) respectively. Thus, \( \sin 40^{\circ} \cos 50^{\circ} + \cos 40^{\circ} \sin 50^{\circ} = \sin(40^{\circ} + 50^{\circ}) \).

Step 3: Simplify the Expression

Calculate the angle inside the sine function: \( 40^{\circ} + 50^{\circ} = 90^{\circ} \). So, the expression simplifies to \( \sin 90^{\circ} \).

Step 4: Evaluate \( \sin 90^{\circ} \)

The value of \( \sin 90^{\circ} \) is a known trigonometric result. Therefore, \( \sin 90^{\circ} = 1 \). This gives us the exact value of the original expression.

Key Concepts

Sine Addition Formula

The Sine Addition Formula is an essential identity in trigonometry which simplifies expressions involving the sine of sums of angles. It is written as \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). This formula emerges when combining two angles' sine values. Understanding this formula allows us to simplify complex trigonometric expressions into more manageable forms.

When you encounter expressions like \( \sin 40^{\circ} \cos 50^{\circ} + \cos 40^{\circ} \sin 50^{\circ} \), recognizing them as the sine of a sum is crucial. Here, we substitute \( a = 40^{\circ} \) and \( b = 50^{\circ} \). Applying the formula, we find that the expression simplifies to \( \sin(40^{\circ} + 50^{\circ}) \). By utilizing this identity, students can not only solve equations faster but also verify solutions through back-calculations.

Exact Trigonometric Values

In trigonometry, certain angle values are recognized as *exact* because their sine, cosine, and tangent values are well-known and do not require a calculator to determine. Angles such as \( 30^{\circ} \), \( 45^{\circ} \), \( 60^{\circ} \), and \( 90^{\circ} \) are examples of these immediate-values.

When simplifying the expression \( \sin(40^{\circ} + 50^{\circ}) \), you calculate the angle to get \( 90^{\circ} \). This angle is special because \( \sin 90^{\circ} \) is one of these exact values, precisely equal to \( 1 \). Knowing these exact values is incredibly helpful in quickly evaluating trigonometric expressions without computational tools. Mastery of these values helps students develop strong intuition across various mathematical problems.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to simpler forms using known identities and values. This skill is beneficial in writing expressions that are more easily interpretable or comparable.

Using known identities like the Sine Addition Formula allows one to simplify initial expressions efficiently. Once the expression \( \sin 40^{\circ} \cos 50^{\circ} + \cos 40^{\circ} \sin 50^{\circ} \) is identified with the sine addition formula, simplification occurs by evaluating \(\sin(90^{\circ})\), leading directly to the solution \( 1 \). This process of simplifying is not only about finding a number but also in showing understanding of deeper mathematical connections within the angles and their trigonometric interpretations.