Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the expression. $$\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}$$

Step-by-Step Solution

Answer

The exact value of the expression \(\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}\) is 0

Step 1: Identify the formula

Identify the applicable formula which is \(\cos(a + b) = \cos a \cos b - \sin a \sin b\), applies here and can be rearranged as \(\cos a \cos b + \sin a \sin b = \cos(a - b)\). The given expression is matching the rearranged form.

Step 2: Substitute the values

Substitute \(a = 120^{\circ}\) and \(b = 30^{\circ}\) into \(\cos(a - b)\) which gives us \(\cos(120^{\circ} - 30^{\circ}) = \cos 90^{\circ}\).

Step 3: Calculate value

The value of \(\cos 90^{\circ}\) is 0 in trigonometric terms. Thus, the given expression \(\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ} = 0\).

Key Concepts

Cosine of Angles

The cosine function is a fundamental concept in trigonometry, representing the ratio of the adjacent side to the hypotenuse in a right triangle. It is a periodic function with a period of 360° or \(2\pi\) radians.

• **Basic Angle Understanding**: For any angle \(\theta\), the cosine value gives us important information about the horizontal component when the angle is placed on the unit circle.
• **Key Cosine Values**: Some important cosine angles to remember include \(\cos 0^{\circ} = 1\), \(\cos 90^{\circ} = 0\), and \(\cos 180^{\circ} = -1\). These values will often appear in problem-solving exercises.

Understanding cosine helps students solve problems involving angle transformations and is essential for interpreting the relationships between angles within geometric figures.

Sine of Angles

Just like cosine, the sine function is one of the basic trigonometric functions. It shows the ratio of the length of the side opposite the angle to the hypotenuse of a right triangle.

• **Periodic Nature**: The sine function is also periodic with a period of 360° or \(2\pi\) radians.
• **Special Angles**: Key angles to remember include \(\sin 0^{\circ} = 0\), \(\sin 90^{\circ} = 1\), and \(\sin 180^{\circ} = 0\).
• **Use in Calculations**: The sine function helps determine vertical components of angles, and it can be used to derive other trigonometric identities when solving problems.

A strong understanding of sine aids in navigating between geometrical and trigonometric realms, especially when angles are presented in degrees or radians.

Angle Subtraction Formula

The angle subtraction formula is a key trigonometric identity that allows for the calculation of the cosine of the difference between two angles. This identity can simplify complex expressions involving trigonometric functions, such as those found in angle transformations.
The formula is written as: \[\cos(a - b) = \cos a \cos b + \sin a \sin b\]

• **Expression Simplification**: It helps convert expressions involving cosine and sine products into a single cosine function, simplifying calculations and interpretation.
• **Real-world Use**: This is particularly helpful when dealing with angles not commonly found on the unit circle, streamlining the process to find unknown trigonometric values.

By mastering the angle subtraction formula, students can tackle a broad range of trigonometric problems, enhancing their mathematical problem-solving skills.