Problem 67 - Algebra 2

Question

Find the value of each function. $$ \cos \left(-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The value is \(\frac{1}{2}\).

Step 1: Identify the Angle

Recognize that the problem involves finding the cosine of the angle \(-\frac{\pi}{3}\). This is in radians and represents a negative angle, indicating a clockwise rotation from the positive x-axis.

Step 2: Use the Even Identity Rule

Recall that cosine is an even function, meaning \(\cos(-x) = \cos(x)\). Thus, \(\cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right)\). We can evaluate the cosine of the positive angle instead.

Step 3: Evaluate \(\cos\left(\frac{\pi}{3}\right)\)

Using the unit circle or a trigonometric table, find that the cosine of \(\frac{\pi}{3}\) is \(\frac{1}{2}\). This is because \(\frac{\pi}{3}\) radians corresponds to 60 degrees, where the cosine value is known to be \(\frac{1}{2}\).

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions and is commonly represented as \(\cos(x)\). It is used to determine the horizontal component of a point on the unit circle that corresponds to a given angle \(x\). Cosine is an even function, which means it exhibits symmetry across the y-axis.
Consequently, \(\cos(-x) = \cos(x)\). This property is extremely useful when evaluating cosine for negative angles, as you simply take the cosine of the positive equivalent.

• On the unit circle, cosine is the x-coordinate of a point.
• When \(\cos(\theta)\) refers to angles in standard position, it helps indicate horizontal distance from the origin.

Knowing these properties makes cosine invaluable in solving equations and understanding wave patterns.

Radians and Degrees

Radians and degrees are two units used to measure angles. While degrees are more commonly used in everyday contexts, radians are often preferred in advanced mathematics because of the natural way they relate angle measures to arc length on the unit circle.

• One full circle is \(360\) degrees or \(2\pi\) radians.
• One radian is the angle created when the arc length equals the radius of the circle.
• Converting between degrees and radians is simple: \(1\,\text{radian} = \frac{180}{\pi}\,\text{degrees}\).

For example, \(\frac{\pi}{3}\) radians converts to \(60\) degrees. Understanding both units allows you to switch between them seamlessly, depending on what each problem requires.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions which remain true for all angles. These identities are essential tools in trigonometry, allowing us to simplify expressions and solve equations effectively.

Even-Odd Identities:

Cosine's role as an even function leads to the identity \(\cos(-x) = \cos(x)\). This is a particular type of identity known as an even-odd identity.

• An **even function** has a symmetrical graph on both sides of the y-axis.
• An **odd function** like sine fulfills \(\sin(-x) = -\sin(x)\).

Pythagorean Identities:

Another fundamental set of identities: For any angle \(\theta\), the Pythagorean identities relate sine and cosine:

• \(\sin^2(x) + \cos^2(x) = 1\)
• \(1 + \tan^2(x) = \sec^2(x)\)
• \(1 + \cot^2(x) = \csc^2(x)\)

By leveraging these identities, complex trigonometric problems become much easier to work through.