Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \cos \left(-30^{\circ}\right) $$

Short answer

\(\cos(-30^{\circ}) = \frac{\sqrt{3}}{2}\)

Step-by-step solution

Identify the Trigonometric Function

Recognize that we need to find the value of \(\cos \left(-30^{\circ}\right)\).

Apply the Even-Odd Property of Cosine

Recall that cosine is an even function, which means \(\cos(-\theta)=\cos(\theta)\). This suggests that \(\cos \left(-30^{\circ}\right) = \cos(30^{\circ})\).

Find the Value of \(\cos(30^{\circ})\)

From trigonometric identities, we know that \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\).

Key concepts

Even-Odd Properties

Trigonometric functions can be classified as either even or odd, based on their symmetry properties. Understanding these properties simplifies computation and helps determine the values of trigonometric functions for negative angles.

An **even function** is symmetric around the y-axis. Mathematically, a function f(x) is even if for every x in its domain, \(f(-x) = f(x)\).

Examples include:

• \(\text{Cosine}\) function: \(\text{cos}(-\theta) = \text{cos}(\theta)\)
• \(\text{Secant}\) function: \(\text{sec}(-\theta) = \text{sec}(\theta)\)

On the other hand, an **odd function** is symmetric around the origin. That is, a function f(x) is odd if for every x in its domain, \(f(-x) = -f(x)\).

Examples include:

• \(\text{Sine}\) function: \(\text{sin}(-\theta) = -\text{sin}(\theta)\)
• \(\text{Tangent}\) function: \(\text{tan}(-\theta) = -\text{tan}(\theta)\)
• \(\text{Cosecant}\) function: \(\text{csc}(-\theta) = -\text{csc}(\theta)\)
• \(\text{Cotangent}\) function: \(\text{cot}(-\theta) = -\text{cot}(\theta)\)

For this problem, we utilize the fact that cosine is an even function to simplify \(\text{cos}(-30^{\text{°}})\) to \(\text{cos}(30^{\text{°}})\).

Cosine Function

The cosine function, denoted as \(\text{cos}(\theta)\), is a fundamental trigonometric function. It relates the angle \(\theta\) with the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Some key properties include:

• **Periodicity**: \(\text{cos}(\theta + 2\text{π}) = \text{cos}(\theta)\)
• **Symmetry (even property)**: \(\text{cos}(-\theta) = \text{cos}(\theta)\)
• **Range**: The values of cosine range from -1 to 1.
• **Amplitude**: The amplitude of the cosine wave is 1.

In this exercise, when finding \(\text{cos}(-30^{\text{°}})\), leveraging the even property helps us recognize that it is the same as \(\text{cos}(30^{\text{°}})\). Therefore, we don't need to recalculate the value.
This makes computations straightforward and avoids mistakes.
In various applications, the cosine function is used to model oscillatory patterns, like sound waves or light waves.

Exact Values

Exact values of trigonometric functions are often derived from special angles, such as \(0^{\text{°}}\), \(30^{\text{°}}\), \(45^{\text{°}}\), \(60^{\text{°}}\), and \(90^{\text{°}}\). These values are essential in solving problems without a calculator.

For the cosine function, some important exact values are:

• \(\text{cos}(0^{\text{°}}) = 1\)
• \(\text{cos}(30^{\text{°}}) = \frac{\text{√3}}{2}\)
• \(\text{cos}(45^{\text{°}}) = \frac{1}{\text{√2}} = \frac{\text{√2}}{2}\)
• \(\text{cos}(60^{\text{°}}) = \frac{1}{2}\)
• \(\text{cos}(90^{\text{°}}) = 0\)

In our problem, we realize that \(\text{cos}(30^{\text{°}})\) is needed. This value is known to be \(\frac{\text{√3}}{2}\).
By understanding these exact values, it is possible to work through trigonometric problems more efficiently and accurately.