Problem 50 - Algebra 2

Question

Find the exact value of each expression. \(\cos \left(-300^{\circ}\right)\)

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Find an Equivalent Positive Angle

To find the cosine of a negative angle, we first find an equivalent positive angle in standard position. Add 360° repeatedly until the angle becomes positive. For \(-300°\), add 360°: \(-300° + 360° = 60°\). So, \(-300°\) is equivalent to \(60°\) in standard position.

Step 2: Use the Cosine of the Equivalent Angle

Now find \(\cos(60°)\). From trigonometric tables or the unit circle, we know that \(\cos(60°) = \frac{1}{2}\).

Key Concepts

Equivalent Angles

When solving trigonometric problems, the concept of equivalent angles is incredibly useful. Equivalent angles are angles that have the same terminal side when positioned in standard form (starting from the positive x-axis). For example, in trigonometry, an angle of \(-300^\circ\) can be challenging to work with, but we can find an equivalent positive angle by adding \(360^\circ\), thus moving it into the first revolution of the unit circle. This gives us:

• \(-300^\circ + 360^\circ = 60^\circ\)

Here, \(-300^\circ\) and \(60^\circ\) are equivalent, meaning they land on the same point on the unit circle. Working with positive angles often simplifies calculations and is commonly used in trigonometric tables. Keep in mind that you can add or subtract multiples of \(360^\circ\) as needed to find other equivalent angles.

Cosine Function

The cosine function is pivotal in trigonometry, describing the cosine of an angle as the x-coordinate of the corresponding point on the unit circle. For an angle \( \theta \), \( \cos(\theta) \) gives us the horizontal distance from the origin to where the angle's arm meets the circle. It allows us to compute distances and angles in right triangles and relate them through:

• Adjacency in a triangle: \(a = \cos(\theta) \times c\)
• Where \(a\) is the length adjacent to the angle, \(c\) is the hypotenuse, and \(\theta\) is the angle

In our example, \(\cos(60^\circ)\) is calculated as \(\frac{1}{2}\), which means the point on the unit circle associated with a \(60^\circ\) angle intersects at the horizontal position of \(0.5\). The cosine function is periodic with a period of \(360^\circ\) or \(2\pi\) radians, repeating its values after every full rotation.

Unit Circle

The unit circle is a crucial foundation to mastering trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate system. The importance comes from the unit circle's ability to simplify understanding of trigonometric functions:

• Each point corresponds to an angle \(\theta\)
• The coordinates of each point \((x, y)\) give \(x = \cos(\theta)\) and \(y = \sin(\theta)\)

For instance, when \(\theta = 60^\circ\), the point on the unit circle is \((\frac{1}{2}, \frac{\sqrt{3}}{2})\), meaning:

• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)

The unit circle is thus fundamental in transforming angles into numeric values for sine and cosine functions, making it a valuable tool for solving a wide array of trigonometric problems.

Negative Angles

Understanding negative angles broadens one's comprehension of the circular nature of trigonometric functions. Angles are measured counterclockwise from the positive x-axis for positive angles, and clockwise for negative ones. To find the trigonometric function values of a negative angle, like \(-300^\circ\), transforming it to an equivalent positive angle simplifies the process:

• Add full rotations (multiples of \(360^\circ\)) to gain a positive measure
• Like moving \(-300^\circ\) to \(60^\circ\) by adding \(360^\circ\)

This means negative angles can be easily interpreted using the usual trigonometric values, as they relate back to their positive counterparts. Negative angles show the symmetry and periodicity of trigonometric functions, which oscillate around zero.