Problem 99 - Precalculus with Limits

Question

In Exercises 99 - 102, use the sum-to-product formulas to find the exact value of the expression. \( \sin 75^\circ + \sin 15^\circ \)

Step-by-Step Solution

Answer

The exact value of \( \sin 75^\circ + \sin 15^\circ \) is \( \frac{\sqrt{6}}{2} \).

Step 1: Identify 'p' and 'q'

First, we identify 'p' and 'q' from the original equation. Here, \(p = 75^\circ\) and \(q = 15^\circ\).

Step 2: Apply the sum-to-product formula

We plug the identified values of p and q into the sum-to-product formula \( \sin p + \sin q = 2 \sin \frac{p + q}{2} \cos \frac{p - q}{2} \). Substituting these values, we get: \(2 \sin \frac{75^\circ + 15^\circ}{2} \cos \frac{75^\circ - 15^\circ}{2} \).

Step 3: Simplify the equation

We next perform the operations inside the sine and cosine functions: \(2 \sin \frac{90^\circ}{2} \cos \frac{60^\circ}{2} \), which simplifies further to \(2 \sin 45^\circ \cos 30^\circ \).

Step 4: Evaluate the sines and cosines

The values we have to substitute are well-known ratios in trigonometry. So, we have: \( 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \), by knowing that \(\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).

Step 5: Calculate the final value

Multiplying the values together, we get the result: \( \frac{\sqrt{6}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and sides of a triangle. They are crucial tools in trigonometry, allowing us to simplify expressions and solve equations involving trigonometric functions. One key category of these identities is the sum-to-product identities. These identities help us transform the sum or difference of trigonometric functions into the product of functions.

In our problem, we use the identity for the sum of two sines:

• \( \sin p + \sin q = 2 \sin \left( \frac{p + q}{2} \right) \cos \left( \frac{p - q}{2} \right) \)

This identity makes it easier to find the exact trigonometric values by converting the expression to a simpler form that involves basic sine and cosine rules.

Understanding and applying these identities can greatly simplify complex trigonometric problems and allow for quicker and more accurate computations. It's like having a secret key to unlock challenging expressions.

Exact Values in Trigonometry

Exact trigonometric values refer to the specific known values of sine, cosine, and tangent for certain angles, usually in degrees like 30°, 45°, 60°, and others. Instead of relying on a calculator for approximate decimal values, these exact ratios are often presented in terms of square roots or fractions.

For instance, when working with \( \sin 45^\circ \) or \( \cos 45^\circ \), we know the exact values are \( \frac{\sqrt{2}}{2} \). Similarly, \( \cos 30^\circ \) gives us \( \frac{\sqrt{3}}{2} \). These values are derived from properties of special right triangles or using the unit circle.

Using these known values is beneficial as it allows solving problems exactly, which is often required in academic settings. It not only increases precision but also helps in understanding the geometric interpretation of sine and cosine as they relate to angles and triangles.

Remember to memorize these common exact values, as they frequently appear in trigonometric problems.

Sine and Cosine Ratios

Sine and cosine are fundamental trigonometric functions that are used to describe relationships in a right triangle, or on the unit circle. They are defined as ratios of the sides of a right triangle, and play a crucial role in the understanding of trigonometry.

The sine of an angle in a right-angled triangle is the ratio of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse. These ratios change depending on the angle, providing a way to express various trigonometric values.

In our example, after simplifying with the sum-to-product identity, we use the ratios:

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

These exact ratios allow us to calculate the expression's value precisely without error. Having this understanding helps in predicting what values to use and when.

Grasping these concepts aids in visualizing trigonometric functions not just as abstract numbers, but as meaningful relationships reflecting angles and distances.