Problem 100 - Precalculus with Limits

Question

In Exercises 99 - 102, use the sum-to-product formulas to find the exact value of the expression. \( \cos 120^\circ + \cos 60^\circ \)

Step-by-Step Solution

Answer

The exact value of the expression \( \cos 120^\circ + \cos 60^\circ \) is 0.

Step 1: Identify A and B

In the given task, let A be \(120^\circ\) (first angle) and B be \(60^\circ\) (second angle).

Step 2: Apply the sum-to-product formula

According to the sum-to-product formula for cosine, \(\cos A + \cos B = 2 \cos \frac{{A + B}}{2} \cos \frac{{A - B}}{2}\). Substitute A with \(120^\circ\) and B with \(60^\circ\). This results in \(2 \cos \frac{{120^\circ + 60^\circ}}{2} \cos \frac{{120^\circ - 60^\circ}}{2}\)

Step 3: Simplify the expression

Simplify each part separately. \(120^\circ + 60^\circ = 180^\circ\) and \(120^\circ - 60^\circ = 60^\circ\). So, the expression becomes \(2 \cos \frac{{180^\circ}}{2} \cos \frac{{60^\circ}}{2}\)

Step 4: Further Simplification

Continue simplifying the expression. It results in \(2 \cos 90^\circ \cos 30^\circ\). The values of \(\cos90^\circ\) and \(\cos30^\circ\) are 0 and \(\sqrt{3}/2\) respectively.

Step 5: Compute the Result

Apply these values into the equation, thus it becomes \(2 * 0 * (\sqrt{3}/2) = 0\)

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is fundamental to geometry and is used in many fields such as physics, engineering, and surveying. The trigonometric functions, which relate the angles of a triangle to the lengths of its sides, include sine, cosine, tangent, and their reciprocals.

When it comes to trigonometry, one critical aspect that students must grasp is the concept of angles, usually measured in degrees or radians. In trigonometric calculations, angles often dictate the values of trigonometric functions. For example, some special angles, like 30°, 45°, and 60°, have known exact values for their sine and cosine. Recognizing these values is crucial for simplifying trigonometric expressions and solving problems.

Exact Values of Trigonometric Functions

In trigonometry, exact value refers to the precise value of a trigonometric function at a specific angle without any approximation. These values are often rooted in the unit circle or special right triangles, such as the 30-60-90 and 45-45-90 triangles.

For instance, within the unit circle, each point on the circumference represents an angle originating at the circle's center, and the coordinates of the point are the values of sine and cosine for that angle. A 30-60-90 triangle, on the other hand, is helpful for remembering exact values for sine and cosine at 30° and 60°. Knowing these values enhances a student's ability to perform calculations more quickly and accurately, which is particularly useful when working with sum-to-product formulas.

The Role of Cosine in Trigonometry

The cosine function is one of the primary trigonometric functions and represents the horizontal coordinate of a point on the unit circle. It is the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This function has a range of values from -1 to 1 and is crucial for solving various trigonometric problems.

Cosine is particularly important when dealing with sum-to-product formulas. These formulas are a set of identities that transform the sum or difference of two cosine functions into a product. This transformation is particularly useful in trigonometric simplification and integration. For example, as illustrated in the exercise solution, knowing that the cosine of 90° is 0 and the cosine of 30° is \(\sqrt{3}/2\) allows us to simplify \(\cos 120° + \cos 60°\) to zero through a sum-to-product formula. These identities and properties help students tackle complex problems with a structured approach.