Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=60^{\circ}, B=60^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \(\cos(120^{\circ})\) is \(-\frac{1}{2}\).

Step 1: Understand the Formula

We need to find \(\cos(A+B)\). The formula for the cosine of the sum of two angles is \(\cos(A+B) = \cos A \cos B - \sin A \sin B\).

Step 2: Plug in the Known Values

Given that \(A = 60^{\circ}\) and \(B = 60^{\circ}\), we plug these values into the formula: \(\cos(60^{\circ}+60^{\circ}) = \cos 60^{\circ} \cos 60^{\circ} - \sin 60^{\circ} \sin 60^{\circ}\).

Step 3: Calculate Trigonometric Functions

Using trigonometric identities, we know \(\cos 60^{\circ} = \frac{1}{2}\) and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). Substitute these into the expression to get: \(\cos(120^{\circ}) = \frac{1}{2} \times \frac{1}{2} - \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}\).

Step 4: Simplify the Expression

Simplify the expression: \(\cos(120^{\circ}) = \frac{1}{4} - \frac{3}{4} = -\frac{2}{4} = -\frac{1}{2}\).

Step 5: Final Answer

The calculation shows that \(\cos(120^{\circ}) = -\frac{1}{2}\).

Key Concepts

Cosine of Sum of Angles

The cosine of the sum of two angles, written as \(\cos(A+B)\), is a fundamental concept in trigonometry. It's a way to determine the cosine of an angle that is the sum of two other angles. The identity is:

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

This formula shows us how to break down the cosine of a summed angle into more manageable parts. Each part is a product of cosines and sines of the individual angles. This makes calculations easier when you're dealing with angles whose trigonometric ratios you already know, like 30°, 45°, or 60°. Remember that this formula only requires knowledge of basic trigonometric values, and a little bit of multiplication and subtraction. Understanding this concept greatly helps in solving many trigonometric problems.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry. They help us understand the relationships between the angles and sides of triangles, especially right-angled ones. The primary trigonometric functions are:

• Cosine (\(\cos\))
• Sine (\(\sin\))
• Tangent (\(\tan\))

For instance, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse, while sine is the opposite side to the hypotenuse.These functions have specific values for key angles such as 30°, 45°, and 60°:

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

Knowing these values is crucial in applying trigonometric identities like the cosine of sum of angles. This foundational knowledge allows you to solve more complex problems by breaking them down into simpler parts.

Angle Sum Formula

The angle sum formula in trigonometry is a powerful tool that helps us find the trigonometric function of the sum of two given angles. In essence, it expands the study of angles beyond the right triangle by allowing us to combine known angles to analyze others.The angle sum formula for cosine is:

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

This formula becomes very handy when you need to compute the cosine of an angle that isn't one of the standard angles. By utilizing the easy-to-remember trigonometric values of simpler angles like those at 30° and 60°, it simplifies the process. For example, if we utilize these known values:

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

We can easily calculate \(\cos(120^{\circ})\) by recognizing 120° as 60° + 60° and applying the angle sum formula. This demonstrates how the angle sum formula provides a structured method to address angular computations.