Problem 18 - Algebra and Trigonometry

Question

\(13-18\) m Use an addition or subtraction formula to write the expression as a trigonometric function of one number, and then find its exact value. $$ \cos \frac{13 \pi}{15} \cos \left(-\frac{\pi}{5}\right)-\sin \frac{13 \pi}{15} \sin \left(-\frac{\pi}{5}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \).

Step 1: Identify the Formula

The expression given resembles the cosine angle addition formula, which is: \( \cos a \cos b - \sin a \sin b = \cos(a+b) \). We will use this to rewrite the expression.

Step 2: Apply the Formula

Recognize that \( a = \frac{13\pi}{15} \) and \( b = -\frac{\pi}{5} \). According to the formula, the expression can be rewritten as \( \cos \left( \frac{13\pi}{15} + \left(-\frac{\pi}{5}\right) \right) \).

Step 3: Simplify the Angle

Calculate the sum of the angles: \( \frac{13\pi}{15} + \left(-\frac{\pi}{5}\right) = \frac{13\pi}{15} - \frac{3\pi}{15} = \frac{10\pi}{15} = \frac{2\pi}{3} \). Thus, the expression simplifies to \( \cos \left( \frac{2\pi}{3} \right) \).

Step 4: Evaluate the Trigonometric Function

Determine the value of \( \cos \left( \frac{2\pi}{3} \right) \). Since \( \frac{2\pi}{3} \) is in the second quadrant where the cosine is negative, the value is \( -\cos \left( \pi - \frac{2\pi}{3} \right) = -\cos \left( \frac{\pi}{3} \right) \). We know \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \), so \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \).

Key Concepts

Angle Addition Formula

The angle addition formula is a key trigonometric identity that helps simplify expressions involving the sum of angles. For cosine, the formula is expressed as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). It effectively combines two angles, \(a\) and \(b\), into a single expression. This transformation makes it much easier to solve trigonometric problems.
Using the angle addition formula, one can turn a complex expression involving multiple trigonometric functions into a simpler cosine expression, like \( \cos(a+b) \). In our example problem, the formula helped simplify the trigonometric expression \( \cos \frac{13 \pi}{15} \cos \left(-\frac{\pi}{5}\right) - \sin \frac{13 \pi}{15} \sin \left(-\frac{\pi}{5}\right) \) into \( \cos \left( \frac{2\pi}{3} \right) \).
The clever use of trigonometric identities like this one is fundamental for more complicated calculations in mathematics, especially when dealing with periodic functions.

Cosine Function

The cosine function is one of the most fundamental trigonometric functions, often symbolized as \( \cos \theta \). It is a measure of an angle's relation to the cosine of the circle. The cosine function is known for its distinct waveform, arising from the adjacent side divided by the hypotenuse in a right triangle.

• Cosine values range between -1 and 1, depending on which quadrant the angle is located in.
• The function is periodic with a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.

In our specific example, the cosine of \( \frac{2\pi}{3} \) needed to be found. This angle is situated in the second quadrant, where the cosine values are negative. Knowing these characteristics helps easily determine that \( \cos \left( \frac{2\pi}{3} \right) = - \frac{1}{2} \). This understanding is vital when calculating the exact value, as it tells us more about the angle's position and its cosine's characteristics.

Exact Trigonometric Values

Determining exact trigonometric values allows us to calculate precise results without approximations. This is especially useful in scenarios where accuracy is crucial, for example in engineering problems or physics simulations.
Exact values are often derived from common angles like \( 0, \frac{\pi}{4}, \frac{\pi}{3}, \) and \( \pi \). These are commonly represented in a unit circle where each angle provides specific trigonometric values. For instance, \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \), and its reflection over the \( y\)-axis, \( \cos \left( \frac{2\pi}{3} \right) \), proves to be \( -\frac{1}{2} \) due to the positioning in different quadrants.
Having a firm grasp on these exact values helps solve problems more efficiently. With practice, you will more readily recall these values, making your computations quicker and more efficient. Recognizing these special angles and their associated trigonometric values is like having a toolbox full of shortcuts at your disposal.