Problem 92 - Algebra and Trigonometry

Question

find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$ \sin \frac{3 \pi}{2} \tan \left(-\frac{8 \pi}{3}\right)+\cos \left(-\frac{5 \pi}{6}\right) $$

Step-by-Step Solution

Answer

- \frac{3\sqrt{3}}{2}

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

The value of \(\sin \frac{3 \pi}{2}\) is -1 because at \(\frac{3 \pi}{2}\) radians, we are at the 'bottom' of the unit circle, where the y-coordinate (which gives the sine value) is -1.

Step 2: Evaluate \(\tan \left(-\frac{8 \pi}{3}\right)\)

Negative angles in trigonometric functions change the direction of rotation but have the same values as their positive counterparts. The period of \(\tan(x)\) is \(\pi\), so \(\tan(-\frac{8\pi}{3}) = \tan(-\frac{8\pi}{3} + \frac{9\pi}{3}) = \tan(\pi/3)\). So the value of \(\tan \left(-\frac{8 \pi}{3}\right)\) would be \(\sqrt{3}\).

Step 3: Evaluate \(\cos \left(-\frac{5\pi}{6}\right)\)

The cosine function is even, meaning that \(\cos(-x) = \cos(x)\). Therefore, \(\cos \left(-\frac{5\pi}{6}\right) = \cos \left(\frac{5\pi}{6}\right)\). At \(\frac{5\pi}{6}\) radians, the x-coordinate of the point on the unit circle (which gives the cosine value) is -\(\sqrt{3}/2\).

Step 4: Add the results together

The final step is simply to apply the original operation to our evaluated trigonometric functions. This gives \(\sin \frac{3 \pi}{2} \tan \left(-\frac{8 \pi}{3}\right) + \cos \left(-\frac{5\pi}{6}\right) = -1*\sqrt{3} - \sqrt{3}/2 = - \frac{3\sqrt{3}}{2}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, defined as a circle with a radius of one unit centered at the origin of a coordinate plane. It's a powerful tool for understanding the properties of trigonometric functions. On this circle, any point can be represented by \((x, y)\), which corresponds to the coordinates \((\text{cos}(\theta), \text{sin}(\theta))\) where \(\theta\) is the angle formed by the line connecting the origin to the point and the positive x-axis.

Using the unit circle, we can find the exact values of trigonometric functions for angles that are multiples of \(\frac{\text{π}}{6}\), \(\frac{\text{π}}{4}\), and \(\frac{\text{π}}{3}\), which are commonly encountered in trigonometry problems. These points on the unit circle are especially helpful when we do not use calculators, as seen in the exercise solution where the value of \(\text{sin}(\frac{3\text{π}}{2})\) is obtained directly from the unit circle's y-coordinate at that angle. The use of the unit circle simplifies complex trigonometric calculations into more manageable visual concepts.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables. They serve as tools to simplify trigonometric expressions and to transform them into other equivalent forms. Some of the most commonly used identities include the Pythagorean identities, the angle sum and difference identities, and the double- and half-angle identities.

For example, the identity \(\text{sin}^2(x) + \text{cos}^2(x) = 1\) is a Pythagorean identity derived from the properties of the unit circle. This particular identity was not directly used in the given exercise, but understanding such identities is essential for solving more complex trigonometric problems. Many trigonometric problems require knowing various identities by heart to quickly recognize the best approach to simplifying an expression or solving an equation.

Periodicity in Trigonometry

Periodicity is a characteristic of certain functions where their values repeat at regular intervals. In the realm of trigonometry, periodicity is a core concept explaining how trigonometric functions behave over intervals known as periods. The main trigonometric functions have well-defined periods: for \(\text{sin}(x)\) and \(\text{cos}(x)\), the period is \(\text{2π}\), whereas for \(\text{tan}(x)\), the period is \(\text{π}\).

In the exercise, recognizing the periodicity of the tangent function helps simplify \(\text{tan}\text{(}{-\frac{8\text{π}}{3}}\text{)}\) to \(\text{tan}\text{(}\frac{\text{π}}{3}\text{)}\) by adding the period \(\text{π}\) to the angle to find a co-terminal angle within the function's principal domain. This understanding of periodicity is crucial to simplify and accurately solve trigonometric expressions without a calculator.

Even and Odd Trigonometric Functions

The concepts of even and odd functions are applicable to trigonometric functions and are used to determine symmetries about the y-axis and the origin, respectively. An even function satisfies the property that \(\text{f}(-x) = \text{f}(x)\), meaning the function's graph is symmetrical about the y-axis. In contrast, an odd function satisfies the property that \(\text{f}(-x) = -\text{f}(x)\), indicating symmetry about the origin.

For instance, the exercise leverages the fact that \(\text{cos}(x)\) is an even function to simplify \(\text{cos}\text{(}{-\frac{5\text{π}}{6}}\text{)}\) to \(\text{cos}\text{(}\frac{5\text{π}}{6}\text{)}\). Understanding the symmetry properties of \(\text{sin}(x)\), \(\text{cos}(x)\), and \(\text{tan}(x)\)—which are odd, even, and odd, respectively—can greatly aid in evaluating trigonometric functions and simplifying expressions, particularly when dealing with negative angles.