Problem 33 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \cos 240^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \(\cos 240^{\circ} = -1/2\)

Step 1: Convert Angle to Sum of Familiar Angles

To find the cosine for \240^{\circ}\, it can be noticed that \240^{\circ}\ can be expressed as \(180^{\circ} + 60^{\circ}\). This makes the calculation easier, as the cosine values for these angles are well-known.

Step 2: Apply Cosine Sum Identity

After the angle \240^{\circ}\ has been expressed in the sum form, the sum identity for cosine can be applied. The cosine sum identity is \(cos(a + b) = cos(a)cos(b) - sin(a)sin(b)\). Replacing \(a\) with \(180^{\circ}\) and \(b\) with \(60^{\circ}\), the equation becomes \(cos(180^{\circ} + 60^{\circ}) = cos(180^{\circ})cos(60^{\circ}) - sin(180^{\circ})sin(60^{\circ})\).

Step 3: Substitute Known Cosine and Sine Values

Now, replacing cos(180°) with -1, cos(60°) with 1/2, sin(180°) with 0, and sin(60°) with √3/2, the equation becomes: \(-1*(1/2) - 0*(√3/2) = -1/2\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that allow us to simplify complex trigonometric expressions. One of the most important identities is the cosine sum identity:

• Cosine Sum Identity: \[\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\]This identity helps in finding the cosine of the sum of two angles by using simpler, more familiar angles.

In the given exercise, the angle \(240^\circ\) is expressed as the sum of \(180^\circ\) and \(60^\circ\). By applying the cosine sum identity, you can easily compute the cosine of this angle using basic trigonometric values. These identities simplify calculations and are especially useful when dealing with non-standard angles.

Exact Trigonometric Values

Exact trigonometric values are specific values of trigonometric functions for standard angles. These values are often memorized as they frequently occur in problems.

• Some key angles include \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\).
• For trigonometric functions like cosine, knowing the exact values for these angles can help solve problems efficiently.

For example, in this exercise, we use:

• \(\cos(180^\circ) = -1\)
• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(180^\circ) = 0\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)

By substituting these exact values into the identities, we can calculate the trigonometric functions of more complex angles such as \(240^\circ\). This process eliminates the need for approximation and yields precise answers, which is valuable for both theoretical and practical applications.

Angle Conversion

Angle conversion is an important skill in trigonometry, involving the transformation of angles into more convenient forms for calculation.

• This often involves breaking down an angle into the sum or difference of more manageable angles.
• In the exercise given, \(240^\circ\) is converted into the sum of \(180^\circ\) and \(60^\circ\).
• Such decompositions utilize familiarity with trigonometric values at standard points, simplifying the calculation process.

This method of angle conversion not only makes use of the cosine sum identity but also demonstrates the power of transforming angles for easier calculation. By recognizing how an angle can be rewritten in terms of known values, you can employ identities more effectively. Such skills are crucial in both academic studies and real-world applications involving trigonometry.