Problem 64 - Algebra 2

Question

Use the fact that \(\frac{\pi}{6}=\frac{\pi}{2}-\frac{\pi}{3}\) to find an exact value for \(\cos \frac{\pi}{6} .\) Show your work.

Step-by-Step Solution

Answer

The exact value for \(\cos \frac{\pi}{6}\) is \(\frac{\sqrt{3}}{2}\).

Step 1: Write the Difference Identity for Cosine

Start by writing the difference identity for cosine which is: \(\cos(a-b) = \cos a \cos b + \sin a \sin b\). We know that \(a = \frac{\pi}{2}\) and \(b = \frac{\pi}{3}\).

Step 2: Substitute \(\frac{\pi}{2}\) and \(\frac{\pi}{3}\) to the Identity

Substitute \(a = \frac{\pi}{2}\) and \(b = \frac{\pi}{3}\) into the formula: \(\cos(\frac{\pi}{2}-\frac{\pi}{3}) = \cos \frac{\pi}{2} \cos \frac{\pi}{3} + \sin \frac{\pi}{2} \sin \frac{\pi}{3}\). As \(\frac{\pi}{2}-\frac{\pi}{3} = \frac{\pi}{6}\), the equation becomes: \(\cos \frac{\pi}{6} = \cos \frac{\pi}{2} \cos \frac{\pi}{3} + \sin \frac{\pi}{2} \sin \frac{\pi}{3}\).

Step 3: Plug in the Known Values of Cos and Sin

Use the known cosine and sine values: \(\cos \frac{\pi}{2} = 0\), \(\cos \frac{\pi}{3} = \frac{1}{2}\), \(\sin \frac{\pi}{2} = 1\), and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). Substitute those into the equation: \(\cos \frac{\pi}{6} = 0 \cdot \frac{1}{2} + 1 \cdot \frac{\sqrt{3}}{2}\).

Step 4: Simplify the Equation

Upon simplification, the equation becomes: \(\cos \frac{\pi}{6} = 0 + \frac{\sqrt{3}}{2}\). Hence, \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)

Key Concepts

Difference Identity for Cosine

The difference identity for cosine is a key formula in trigonometry that allows us to find the cosine of an angle expressed as the difference of two known angles. The formula is written as:

• \[\cos(a - b) = \cos a \cos b + \sin a \sin b\]

This identity breaks down the cosine of a difference into easier, manageable parts by using cosine and sine values of the individual angles. Understanding this identity is crucial for solving many trigonometric problems where angles are not immediately simple.
To apply the difference identity accurately, it’s important to identify the correct angles for substitution and to remember the trigonometric values for these basic angles, which can then be substituted into the identity to solve for the desired cosine value.

Exact Values for Trigonometric Functions

In trigonometry, the term 'exact values' refers to the known precise values of trigonometric functions like sine, cosine, and tangent for specific angles. These values often involve simple fractions or radicals since they are derived from geometry, particularly the properties of right-angle triangles and unit circles.
These exact values are foundational. They help solve trigonometric problems without using a calculator. Some commonly known exact values include:

• \( \cos \frac{\pi}{2} = 0 \)
• \( \cos \frac{\pi}{3} = \frac{1}{2} \)
• \( \sin \frac{\pi}{2} = 1 \)
• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)

Knowing these values is essential, as they turn up frequently in exercises concerning trigonometric identities and calculations. Using such known values, we can work through identities like the difference identity for cosine to find unknown trigonometric function values.

Angle Subtraction Formulas

Angle subtraction formulas are a collection of identities in trigonometry that express trigonometric functions of angle differences in terms of the functions of the angles themselves. These formulas are especially useful when working with non-standard angles that can be expressed as differences of standard angles.
The cosine subtraction formula, for example, is one such identity:

• \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]

By substituting known angle values into this formula, we can calculate the cosine for angles like \[ \frac{\pi}{6} = \frac{\pi}{2} - \frac{\pi}{3} \]. It's important to practice these formulas and understand how they transform complex trigonometric expressions into simpler ones. Knowing these formulas by heart allows for reliable manipulation and simplification of trigonometric expressions to derive exact values.