Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \cos \left(-270^{\circ}\right) $$

Short answer

The exact value is \(\text{cos}(-270^{\text{o}}) = 0\).

Step-by-step solution

Understand the Even-Odd Properties

An important property of cosine is that it is an even function, which means \(\text{cos}(-x) = \text{cos}(x)\). This property will help in solving the given expression.

Apply the Even-Odd Property

Since \(\text{cos}(-x) = \text{cos}(x)\), we can say \(\text{cos}(-270^{\text{o}}) = \text{cos}(270^{\text{o}})\).

Determine the Reference Angle

The reference angle for \(270^{\text{o}}\) is \(270^{\text{o}} - 180^{\text{o}} = 90^{\text{o}}\). However, since \(270^{\text{o}}\) is in standard position, \(270^{\text{o}}\) directly indicates it is on the negative y-axis.

Apply the Unit Circle Concept

At \(270^{\text{o}}\), the coordinates on the unit circle are \( (0, -1) \). The cosine of an angle corresponds to the x-coordinate of its point on the unit circle.

Find the Cosine Value

Thus, the x-coordinate at \(270^{\text{o}}\) is \(0\). Therefore, \(\text{cos}(270^{\text{o}}) = 0\).

Key concepts

even function

Understanding the concept of even functions is essential in trigonometry, particularly when working with the cosine function. An even function is one that satisfies the condition \(f(-x) = f(x)\) for all x in its domain. The cosine function, \(\cos(x)\), is a perfect example of an even function. This property can simplify calculations, especially when dealing with negative angles.
Consider the problem where you need to find \(\cos(-270^{\circ})\). Using the even function property, we know:
\[ \cos(-270^{\circ}) = \cos(270^{\circ}) \]
This property eliminates the need to work with the negative angle directly. Instead, we simply evaluate the cosine of the positive angle.

cosine

Cosine is one of the primary trigonometric functions, often abbreviated as \(\cos(x)\). It helps us relate the angles to the coordinates of points on the unit circle. For any angle \(x\), \(\cos(x)\) gives the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
In our example, we applied the even function property to reduce \(\cos(-270^{\circ})\) to \(\cos(270^{\circ})\).
Once we identify that we need \(\cos(270^{\circ})\), we can use the unit circle to find the exact value. The unit circle plays a critical role in trigonometry, linking angles to their sine and cosine values.

unit circle

The unit circle is a powerful tool in trigonometry. It's a circle centered at the origin \((0, 0)\) with a radius of 1. On the unit circle, any point \( (x, y) \) represents \(\cos(\theta)\) and \(\sin(\theta)\) for a given angle \(\theta\) measured from the positive x-axis.
In our specific problem, when looking at \(270^{\circ}\), it corresponds to the point directly on the negative y-axis. This point is \( (0, -1) \).
Since the cosine of an angle is the x-coordinate of its corresponding point on the unit circle:

• The x-coordinate at \(270^{\circ}\) is 0.
• This means \(\cos(270^{\circ}) = 0\).

Understanding the unit circle helps us quickly determine these values without the need for extensive calculations.