Problem 32 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\cos \left(-30^{\circ}\right)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Negative Angle Property

The cosine function is an even function, which means that \( \cos(-\theta) = \cos(\theta) \). This property tells us that when calculating the cosine of a negative angle, the result is the same as calculating the cosine of the positive angle.

Step 2: Use the Property to Simplify

Apply the even function property to simplify the problem: \( \cos(-30^{\circ}) = \cos(30^{\circ}) \). Now, instead of calculating \( \cos(-30^{\circ}) \), we need to calculate \( \cos(30^{\circ}) \).

Step 3: Recall the Exact Value of \( \cos(30^{\circ}) \)

Use your knowledge of the unit circle or trigonometric ratios to find \( \cos(30^{\circ}) \). The exact value of \( \cos(30^{\circ}) \) is \( \frac{\sqrt{3}}{2} \), because \( 30^{\circ} \) corresponds to \( \frac{\pi}{6} \) radians, and it's a standard angle on the unit circle.

Key Concepts

even function property

In mathematics, certain functions like the cosine function have special attributes. The cosine function is known as an even function. This means it has symmetry around the y-axis. When we say a function is even, it implies that it satisfies the condition \( f(-x) = f(x) \) for every \( x \) in the domain of the function. For cosine, this is expressed as \( \cos(-\theta) = \cos(\theta) \).

This "even function property" of cosine is particularly useful when dealing with negative angles because it allows us to transform the problem easily. When you are given \( \cos(-30^{\circ}) \), you can use the property to know it equals \( \cos(30^{\circ}) \) without further calculations. Such properties help simplify our work and prevent potential errors.

unit circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one that is centered at the origin of a coordinate plane. The beauty of the unit circle is that it allows you to define trigonometric functions for all real numbers.

• Every angle \( \theta \) on the unit circle corresponds to a point \((x, y)\).
• The x-coordinate of this point is \( \cos(\theta) \), and the y-coordinate is \( \sin(\theta) \).
• The radius, being 1, makes calculations involved with the circle easier.

For example, when finding \( \cos(30^{\circ}) \), you reference the unit circle where a 30-degree angle corresponds to \( \frac{\pi}{6} \) radians. Here, the cosine value, which is the x-coordinate on the unit circle, is \( \frac{\sqrt{3}}{2} \).

Using the unit circle, you quickly navigate standard angles like 30°, 45°, and 60° and find their cosine and sine values without difficulty.

trigonometric ratios

Trigonometric ratios link angles to their respective sides in a right-angled triangle. They are foundational to trigonometry, and involve three primary functions: sine (sin), cosine (cos), and tangent (tan). These functions correspond to ratios of sides in a triangle.

• \( \cos \) relates the adjacent side to the hypotenuse.
• \( \sin \) is the opposite side over the hypotenuse.
• \( \tan \) is the opposite side over the adjacent side and is the ratio of \( \sin \) to \( \cos \).

For \( 30^{\circ} \), using the trigonometric ratios helps determine that \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \).

When you use trigonometric ratios, particularly those derived from a 30-60-90 triangle, you recognize these predictable ratios and apply them across problems. This makes remembering exact values like \( \cos(30^{\circ}) \) easy and reliable.