Problem 4 - Algebra and Trigonometry

Question

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Find the exact value of each expression. $$ \cos \left(\frac{2 \pi}{3}-\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

The exact value of \( \cos \left(\frac{2 \pi}{3}-\frac{\pi}{6}\right) \) is 0.

Step 1: Identify the Values of a and b

Here, \( a = \frac{2 \pi}{3} \) and \( b = \frac{\pi}{6} \). These are the two angles whose difference is given to us.

Step 2: Apply the Formula

Now, we apply the formula \( \cos (a - b) = \cos a \cos b + \sin a \sin b \). Plugging the values of \( a \) and \( b \) into the equation, we get: \( \cos \left(\frac{2 \pi}{3}-\frac{\pi}{6}\right) = \cos \frac{2 \pi}{3} \cos \frac{\pi}{6} + \sin \frac{2 \pi}{3} \sin \frac{\pi}{6}\).

Step 3: Evaluate the Trigonometric Values

In this step, evaluate each trigonometric function at the given angles. We know that \( \cos \frac{2 \pi}{3} = -\frac{1}{2} \), \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), \( \sin \frac{2 \pi}{3} = \frac{\sqrt{3}}{2} \), and \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 4: Substitute the Values

Substitute the values from the previous step into the formula: \( -\frac{1}{2} * \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}* \frac{1}{2} = -\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} \).

Step 5: Calculate the Result

In the final step, calculate the result: \( -\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = 0 \). So, \( \cos \left(\frac{2 \pi}{3}-\frac{\pi}{6}\right) = 0 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. These identities are helpful tools in simplifying and solving trigonometric expressions and equations. When dealing with problems like the cosine of the difference between two angles, we use the angle subtraction identity. Specifically, for cosine, the identity is given by:\[ \cos (a - b) = \cos a \cos b + \sin a \sin b \]This formula allows us to express the cosine of a difference of two angles as a function of the sines and cosines of the individual angles. By using this identity, we can break down complex trigonometric expressions into simpler parts, making calculations easier.

Exact Trigonometric Values

Exact trigonometric values refer to the specific known values of sine, cosine, and tangent for certain standard angles, such as 30°, 45°, 60°, and their radian equivalents. These values are often used as benchmarks in solving trigonometric problems without using a calculator.Here are some key exact values:

• For \( \pi/6 \) (30°), we have \( \cos \pi/6 = \frac{\sqrt{3}}{2} \) and \( \sin \pi/6 = \frac{1}{2} \),
• For \( 2\pi/3 \) (120°), we find \( \cos 2\pi/3 = -\frac{1}{2} \) and \( \sin 2\pi/3 = \frac{\sqrt{3}}{2} \).

These values are derived from the unit circle, providing precise measures that are essential for accuracy in trigonometric calculations.

Angle Subtraction in Trigonometry

Angle subtraction in trigonometry involves calculating the sine, cosine, or tangent of the difference between two angles. This concept is crucial when dealing with expressions like \( \cos(\frac{2\pi}{3} - \frac{\pi}{6}) \). Using identities for subtraction can transform a complex expression into simpler, more manageable components.The key here is to:

• Identify the two angles involved,
• Apply the appropriate identity, such as the cosine difference identity mentioned earlier, and
• Utilize known exact trigonometric values to evaluate the expression.

Mastering angle subtraction helps in understanding more intricate trigonometric expressions and proofs, offering a solid foundation for analytical problem-solving in mathematics.