Problem 91 - Algebra and Trigonometry

Question

find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$ \sin \frac{3 \pi}{2} \tan \left(-\frac{15 \pi}{4}\right)-\cos \left(-\frac{5 \pi}{3}\right) $$

Step-by-Step Solution

Answer

-3/2

Step 1: Evaluate \(\sin \frac{3 \pi}{2}\)

Since sine represents the y-coordinate on the unit circle, and \(\frac{3 \pi}{2}\) corresponds to the point (0,-1), so \(\sin \frac{3 \pi}{2}\) equals -1.

Step 2: Evaluate \(\tan\left(-\frac{15 \pi}{4}\right)\)

Remember, \(\tan \theta = \sin \theta / \cos \theta\). Because tangent has a period of \(\pi\), add \(\pi\) to the angle \(-\frac{15 \pi}{4}\) until you get an angle that you know. The angle \(-\frac{15 \pi}{4}\) is equivalent to \(\frac{ \pi}{4}\). Thus, \(\tan\left(-\frac{15 \pi}{4}\right)\) equals 1.

Step 3: Multiply the results of step 1 and step 2

Multiply \(\sin \frac{3 \pi}{2}\) and \(\tan\left(-\frac{15 \pi}{4}\right)\). The result is -1.

Step 4: Evaluate \(\cos\left(-\frac{5 \pi}{3}\right)\)

Cosine has a period of \(2\pi\) and represents the x-coordinate on the unit circle. Based on symmetry, \(\cos\left(-\frac{5 \pi}{3}\right)\) is equivalent to \(\cos\left(\frac{\pi}{3}\right)\), which is equal to 1/2.

Step 5: Subtract the results of step 3 and step 4

Subtract \(\cos\left(-\frac{5 \pi}{3}\right)\) from the result of Step 3: -1 - 1/2 = -3/2.

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry that helps us easily find values of trigonometric functions for various angles. By definition, the unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It allows us to visualize how sine and cosine behave as they trace the circle.

One of the key aspects of the unit circle is its standardized points, which represent notable angles such as \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \), among others. When working with these points:

• The x-coordinate denotes the cosine of the angle.
• The y-coordinate indicates the sine of the angle.

For instance, the angle \( \frac{3\pi}{2} \) corresponds to the point (0, -1) on the unit circle, where the sine value is -1, and the cosine is 0. This insight can simplify complex trigonometric problems by mapping challenging angles to more familiar counterparts using the unit circle as a reference.

Periodicity of Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, exhibit periodic behavior that allows us to predict their values over repeating intervals.

This principle is particularly useful when dealing with angles larger than \( 2\pi \) or negative angles, as we can reduce them to equivalent angles within the primary cycle.

• The sine and cosine functions both have a period of \( 2\pi \). This means that \( \sin(\theta + 2k\pi) = \sin(\theta) \) and \( \cos(\theta + 2k\pi) = \cos(\theta) \) for any integer \( k \).
• The tangent function, in contrast, has a period of \( \pi \). Thus, \( \tan(\theta + k\pi) = \tan(\theta) \).

In our exercise, \( \tan(-\frac{15\pi}{4}) \) simplifies to \( \frac{\pi}{4} \) because tangent’s period of \( \pi \) allows us to add multiples of \( \pi \) to reduce complex angles.
Understanding periodicity helps us work with trigonometric functions more flexibly and uncover simpler forms of complicated expressions.

Exact Values of Trigonometric Functions

Knowing the exact values of trigonometric functions for specific angles is crucial for solving trigonometry problems efficiently. These exact values are typically derived from the unit circle and commonly known special triangles, such as the 30°-60°-90° and 45°-45°-90° triangles.

Common angles with exact trigonometric values include:

• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \cos \frac{\pi}{3} = \frac{1}{2} \)

In the problem, knowing that \( \cos(-\frac{5\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2} \) allowed us to identify this exact value. Identifying these trigonometric values without a calculator saves time and helps ensure accuracy in both study and exams.