Problem 7 - Algebra 2

Question

Use a double-angle identity to find the exact value of each expression. $$ \cos 600^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 600^{\circ} \) is -1/2.

Step 1: Simplify the Angle

Since cosine is periodic with period 360, then \( \cos(600^{\circ}) = \cos(600^{\circ} - 2*360^{\circ}) = \cos(-120^{\circ}) \). It's evident that the angle is negative, but the cosine function is even, meaning \( \cos(x) = \cos(-x) \), so \( \cos(-120^{\circ}) = \cos(120^{\circ}) \).

Step 2: Apply Double Angle Identity

Now, express the given angle in terms of a double-angle identity. One way to do this is by using the identity \( \cos(120^{\circ}) = \cos(2*60^{\circ}) \). Use double angle identity for cosine \( \cos(2A) = 1 - 2\sin^2(A) \). Substituting 60 degrees into the formula gives \( \cos(120^{\circ}) = 1 - 2\sin^2(60^{\circ}) \).

Step 3: Calculate the Exact Value

Use the unit circle to find the sine of 60 degrees, which is \( \sqrt{3}/2 \). Substituting this value into the identity gives \( \cos(120^{\circ}) = 1 - 2*(\sqrt{3}/2)^2 = 1 - 2*(3/4) = -1/2 \).

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry, widely used in mathematics to study the properties of angles in various geometric contexts. The cosine function, often denoted as \( \cos \), gives the horizontal coordinate of a point on a unit circle corresponding to a given angle. In practical terms:

• The cosine of an angle can be seen as the adjacent side over the hypotenuse in a right triangle.
• It relates directly to the unit circle, which simplifies understanding trigonometric values.
• The function is essential for calculating projections along a specified axis.

When working with angles, the cosine function has some distinct properties. It is an even function, meaning that \( \cos(x) = \cos(-x) \), which is incredibly useful in simplifying expressions involving negative angles. Also, when dealing with angle transformations, identities like the double-angle identity \( \cos(2A) = 1 - 2\sin^2(A) \) aid in rewriting complex expressions.

Periodic Function

Functions that repeat their values at regular intervals are known as periodic functions, and the cosine function is a classic example. It has a period of \( 360^{\circ} \) or \( 2\pi \) radians. This periodicity means:

• The function returns the same value every full rotation around the unit circle.
• Knowing the period helps in reducing angles - for example, \( \cos(600^{\circ}) \) can simplify to \( \cos(240^{\circ}) \) after subtracting multiples of 360 degrees.
• This also illustrates that trigonometric functions can model cyclical phenomena such as waves or harmonic motions.

Periodic functions like cosine help in determining equivalent angles, making complex calculations easier by exploiting their repetitive nature to simplify expressions.

Unit Circle

The unit circle is an invaluable tool in trigonometry, offering a geometric representation of the cosine and sine of angles.

• It is defined as a circle with a radius of one centered at the origin of a coordinate plane.
• Angles are typically measured from the positive x-axis, moving counterclockwise.
• For any angle on the unit circle, the cosine value corresponds to the x-coordinate, while the sine corresponds to the y-coordinate.

In the context of our exercise, the unit circle is pivotal for determining \( \sin(60^{\circ}) = \sqrt{3}/2 \), which is used in the double-angle identity to find \( \cos(120^{\circ}) \). This geometric approach simplifies understanding and guides the manipulation of trigonometric functions.