Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \csc \left(-\frac{\pi}{3}\right) $$

Short answer

-\frac{2\text{√3}}{3}

Step-by-step solution

Identify the Even-Odd Property of Cosecant

Cosecant is the reciprocal of the sine function. Since \(\text{sin}(-x) = -\text{sin}(x)\), the cosecant function also obeys this odd property: \(\text{csc}(-x) = -\text{csc}(x)\).

Apply the Property to the Given Angle

Given \(\text{csc}(-\frac{\text{π}}{3})\), we can use the property established in Step 1: \(\text{csc}(-\frac{\text{π}}{3}) = -\text{csc}(\frac{\text{π}}{3})\).

Find \(\text{csc}(\frac{\text{π}}{3})\) Using Known Values

\(\text{csc}(\frac{\text{π}}{3})\) is the reciprocal of \(\text{sin}(\frac{\text{π}}{3})\). Since \(\text{sin}(\frac{\text{π}}{3}) = \frac{\text{√3}}{2}\), this means \(\text{csc}(\frac{\text{π}}{3}) = \frac{2}{\text{√3}}\).

Simplify the Reciprocal Value

Simplify \(\frac{2}{\text{√3}}\). To rationalize the denominator, multiply the numerator and denominator by \(\text{√3}\), resulting in \(\frac{2\text{√3}}{3}\).

Combine the Results

Using the result from Step 2, we get: \(\text{csc}(-\frac{\text{π}}{3}) = -\frac{2\text{√3}}{3}\).

Key concepts

cosecant function

The cosecant function, often denoted as \(\text{csc}\), is one of the six fundamental trigonometric functions. It’s the reciprocal of the sine function. Expressed mathematically: \[ \text{csc}(x) = \frac{1}{\text{sin}(x)} \]
Its domain excludes multiples of \(\text{π}\), where the sine function is zero, since division by zero is undefined. Understanding the cosecant function is critical for solving trigonometric problems, because it ties directly into reciprocal identities and the properties of angles.
Furthermore, knowing the exact values of cosecant for special angles like \(\frac{\text{π}}{6}\), \(\frac{\text{π}}{4}\), and \(\frac{\text{π}}{3}\) is useful across many areas of mathematics and physics.

reciprocal trigonometric identities

Reciprocal trigonometric identities are fundamental relationships between the sine, cosine, and tangent functions and their counterparts: cosecant, secant, and cotangent. These identities help in simplifying complex trigonometric expressions.

• \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\)
• \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\)
• \(\text{cot}(x) = \frac{1}{\text{tan}(x)}\)

In the exercise, we used the fact that \(\text{csc}(x)\) is the reciprocal of \(\text{sin}(x)\). We found \(\text{csc}(\frac{\text{π}}{3})\) by taking the reciprocal of \(\text{sin}(\frac{\text{π}}{3}) = \frac{\text{√3}}{2}\), resulting in \(\text{csc}(\frac{\text{π}}{3}) = \frac{2}{\text{√3}}\). Knowing these relationships simplifies solving trigonometric equations and understanding angle properties.

angle properties in trigonometry

Angles in trigonometry have inherent properties that help in solving trigonometric functions accurately. Each trigonometric function has specific behavior when applied to negative angles or angles outside the standard interval of \([0, 2\text{π}]\).When dealing with negative angles, it’s essential to remember the even-odd properties of trigonometric functions:

• Sine and tangent are odd: \(\text{sin}(-x) = -\text{sin}(x)\) and \(\text{tan}(-x) = -\text{tan}(x)\)
• Cosine is even: \(\text{cos}(-x) = \text{cos}(x)\)

In our solution, the property \(\text{csc}(-x) = -\text{csc}(x)\) played a crucial role, showing its odd nature similar to sine.

rationalizing denominators in trigonometry

Rationalizing the denominator is a process to eliminate roots from the bottom of a fraction. It often involves multiplying the numerator and the denominator by a conjugate or an appropriate root.
In this exercise, after finding \(\text{csc}(\frac{\text{π}}{3}) = \frac{2}{\text{√3}}\), we rationalized the denominator by multiplying by \(\text{√3}/\text{√3}\): \[ \frac{2}{\text{√3}} \times \frac{\text{√3}}{\text{√3}} = \frac{2\text{√3}}{3} \]. This step ensures the final expression is simplified and aesthetically standard.
Rationalizing is essential in algebra and trigonometry, where simplified expressions make subsequent calculations cleaner and more manageable.