Problem 90 - Algebra and Trigonometry

Question

find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$ \sin \frac{17 \pi}{3} \cos \frac{5 \pi}{4}+\cos \frac{17 \pi}{3} \sin \frac{5 \pi}{4} $$

Step-by-Step Solution

Answer

The exact value is \(-(\sqrt{6} + \sqrt{2})/4\).

Step 1: Simplify the arguments of the sine and cosine into standard position

Sine and cosine functions are periodic functions with a period of \(2\pi\). We can subtract or add any multiple of \(2\pi\) to simplify the arguments of the sine and cosine. For \(\frac{17 \pi}{3}\), subtract \(\frac{16 \pi}{3}\) to get \(\pi/3\). And for \(\frac{5 \pi}{4}\), subtract \(2\pi = \frac{8 \pi}{4}\) to get \(-\frac{3 \pi}{4}\).

Step 2: Substitute simplified arguments back into expression

The original expression simplifies to \( \sin \frac{\pi}{3} \cos -\frac{3\pi}{4}+\cos \frac{\pi}{3} \sin -\frac{3\pi}{4}\).

Step 3: Apply the sine and cosine of special angles

From the unit circle, we know \(sin(\frac{\pi}{3}) = \sqrt{3}/2\), \(cos(\frac{\pi}{3}) = 1/2\), \(cos(-\frac{3\pi}{4}) = -1/\sqrt{2}\) and \(sin(-\frac{3\pi}{4}) = -1/\sqrt{2}\). Substituting these into the above expression we get \((\sqrt{3}/2) * (-1/\sqrt{2}) + (1/2) * (-1/\sqrt{2}) = -\sqrt{3}/2\sqrt{2} - 1/2\sqrt{2}\).

Step 4: Rationalize the denominators and simplify the expression

Multiply the numerator and denominators by \(\sqrt{2}\) to rationalize, we get \(-\sqrt{6}/4 - \sqrt{2}/4 = -(\sqrt{6} + \sqrt{2})/4\).

Key Concepts

Periodic Functions

Trigonometric functions like sine and cosine are periodic, meaning they repeat values at regular intervals. The period of these functions is typically

• 2π: This means after every full rotation of the circle, the sine and cosine values repeat.

In the given exercise, we simplify angles by reducing them using multiples of 2π. For instance, converting \(\frac{17π}{3}\) by subtracting \(2π = \frac{16π}{3}\)to get it down to \(\frac{π}{3}\). This simplification helps us deal with smaller angles that lie within one cycle of the circle for easier calculation.

Special Angles

Special angles refer to angles like

• π/3, π/4, π/6: These are common angles often seen in trigonometry problems.

Recognizing these angles is important because they have well-known sine and cosine values. In our exercise:

• sin(π/3) = \(\sqrt{3}/2\)
• cos(π/3) = 1/2
• cos(-3π/4) = -1/\(\sqrt{2}\)
• sin(-3π/4) = -1/\(\sqrt{2}\)

These known values help us calculate the expression accurately without needing a calculator.

Unit Circle

The unit circle is a crucial tool in trigonometry, used to find angle values for sine and cosine functions.

• Radius of 1: The circle has a radius of 1, making calculations straightforward.
• Coordinates: Each point on the circle corresponds to a coordinate \((\cos θ, \sin θ)\), where different angles lead to predictable points on the circle.

For example:

• π/3: Located in the first quadrant with coordinates \((1/2, \sqrt{3}/2)\).
• -3π/4: Found in the third quadrant with coordinates \((-\sqrt{2}/2, -\sqrt{2}/2)\).

This helps in deriving the values used in solving the expression in the exercise.

Rationalizing Denominators

Rationalizing involves converting an irrational denominator into a rational number. This is common in trigonometric problems where roots appear in the denominator.

• Multiply by Conjugate: To rationalize \(\frac{x}{\sqrt{2}}\), multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\).

In the exercise, expressions \(-\sqrt{3}/2\sqrt{2}\) and \(-1/2\sqrt{2}\) are rationalized to yield\[-\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)\]This not only simplifies the expression but also ensures that denominators are free from roots, which is often considered a more "finished" or cleaner form in mathematics.