Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin 76^{\circ} \cos 31^{\circ}-\cos 76^{\circ} \sin 31^{\circ}$$

Step-by-Step Solution

Answer

The exact value is \( \frac{\sqrt{2}}{2} \).

Step 1: Identify the Trigonometric Identity

Recognize that the expression \( \sin 76^{\circ} \cos 31^{\circ} - \cos 76^{\circ} \sin 31^{\circ} \) matches the form of the sine subtraction identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).

Step 2: Apply the Sine Identity

Apply the identity to the expression by setting \( A = 76^{\circ} \) and \( B = 31^{\circ} \). Thus, we have \( \sin(76^{\circ} - 31^{\circ}) \).

Step 3: Calculate the Resulting Angle

Subtract the angle \( B \) from angle \( A \): \( 76^{\circ} - 31^{\circ} = 45^{\circ} \). This simplifies the expression to \( \sin 45^{\circ} \).

Step 4: Determine the Exact Value

Recall that \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). Use this identity to solve the expression.

Key Concepts

Sine Subtraction Identity

In trigonometry, identities are essential tools that help simplify expressions and solve equations. One such identity is the Sine Subtraction Identity, which allows us to rewrite the difference of two angles in terms of sine and cosine products. This identity states:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

Whenever you see an expression that matches this form, you can replace it with \( \sin(A - B) \). This powerful tool can transform complex expressions into something much simpler by recognizing and applying the identity appropriately.
In the given exercise, the expression \( \sin 76^{\circ} \cos 31^{\circ} - \cos 76^{\circ} \sin 31^{\circ} \) perfectly fits the sine subtraction pattern, confirming its transformation into \( \sin(76^{\circ} - 31^{\circ}) \). Understanding and using this identity can significantly reduce computational effort and enhance problem-solving efficiency.

Exact Values of Trigonometric Functions

Finding the exact values of trigonometric functions is a crucial skill in trigonometry, enabling us to work with angles and functions precisely without relying on calculators. This method involves memorizing key angle values that appear frequently in problems, such as 0°, 30°, 45°, 60°, and 90°, among others.
For example, the sine and cosine of 45° are both \( \frac{\sqrt{2}}{2} \). This symmetrical property makes 45° a special angle in trigonometry. Recognizing these values is important as it allows us to solve expressions accurately without estimation or digital aids.
In contexts such as the current exercise, effectively using exact values provides a direct path to the solution once the expression is simplified using identities. After applying the sine subtraction identity in our problem, we arrived at \( \sin 45^{\circ} \), leading directly to the known exact value of \( \frac{\sqrt{2}}{2} \). Understanding these core angles and their trigonometric values is fundamental for solving a wide range of mathematical problems efficiently.

Angle Calculation

Angle calculation is often a straightforward step but requires careful attention to detail, particularly when applying trigonometric identities. It involves subtracting, adding, or otherwise manipulating angles within an expression to simplify it or prepare it for further calculations.
In the given task, once the expression is recognized as fitting the sine subtraction identity, calculating the angle becomes as simple as subtracting the smaller angle from the larger one. Here, subtracting \( 31^{\circ} \) from \( 76^{\circ} \) results in \( 45^{\circ} \). This subtraction is a key intermediate step that leads to a simplified expression that can easily be evaluated using known exact values.
By understanding and performing angle calculations correctly, you'll ensure that trigonometric identities are applied accurately, leading to correct solutions. This foundational skill is applicable in more complex problems and is crucial for mastering trigonometric identities and their applications.