Problem 39 - Precalculus with Limits

Question

In Exercises 37 - 42, find the exact value of the expression. \( \sin 120^\circ \cos 60^\circ - \cos 120^\circ \sin 60^\circ \)

Step-by-Step Solution

Answer

The exact value of the expression is \( \sqrt{3}/2 \).

Step 1: Identify the identity to use

For the given expression \( \sin 120^\circ \cos 60^\circ - \cos 120^\circ \sin 60^\circ \), it can be identified that this is in the form of sine of difference of two angles \( \sin(A - B) = \sin A \cos B - \cos A \sin B \), where A is 120 degrees and B is 60 degrees.

Step 2: Apply the identity

By applying the identity, the above expression can be rewritten as \( \sin(120^\circ - 60^\circ) = \sin 60^\circ \).

Step 3: Calculate the value

We know that \( \sin 60^\circ = \sqrt{3}/2 \). So, the value of the given expression is \( \sqrt{3}/2 \).

Key Concepts

Sine of Difference Formula

To solve trigonometric expressions like \( \sin 120^\circ \cos 60^\circ - \cos 120^\circ \sin 60^\circ \), we use the sine of difference formula. This formula is a key trigonometric identity that helps simplify expressions involving the sine of two angles.

The sine of difference formula is given by:

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

In this formula, \( A \) and \( B \) are angles.
The formula states that the sine of the difference between two angles is equal to the difference of the products of sines and cosines of these angles.

It allows us to break down complex trigonometric expressions into simpler forms. In our problem, \( A \) is \( 120^\circ \) and \( B \) is \( 60^\circ \), which fits perfectly into the sine of difference format. This allows us to directly calculate the sine of a new angle, \( 60^\circ \), leading to a simpler solution.

Exact Values of Trigonometric Functions

In trigonometry, the exact values of certain functions are essential for solving problems accurately. These values are typically based on angles like \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), \( 90^\circ \), and so on.

For example:

• \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
• \( \cos 60^\circ = \frac{1}{2} \)
• \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \cos 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \sin 30^\circ = \frac{1}{2} \)
• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)

These values are derived based on the unit circle and special right triangles, such as the 30-60-90 and 45-45-90 triangles.

Knowing and using these exact values makes evaluating trigonometric expressions simpler and more precise, as seen in our problem where \( \sin 60^\circ = \frac{\sqrt{3}}{2} \) is essential for finding the solution.

Angle Subtraction Identities

Angle subtraction identities are crucial in trigonometry since they allow the simplification of expressions where complex angles are involved. These identities help express functions like sine, cosine, and tangent in terms of more coordinate angles.

For example, the sine of angle subtraction identity is given as:

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

Similarly, for cosine and tangent, we have:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
• \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

These identities are incredibly useful, particularly when dealing with angles that are not standard, like \( 105^\circ \), \( 135^\circ \), etc., allowing them to be broken into smaller angles we know the exact values of.

In our example, using the angle subtraction identity for sine simplifies \( \sin 120^\circ \cos 60^\circ - \cos 120^\circ \sin 60^\circ \) to \( \sin 60^\circ \), which is straightforward to evaluate.