Problem 12 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\cos \left(-60^{\circ}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \cos(-60^\circ) \) is \( \frac{1}{2} \).

Step 1: Determine Angle in Standard Position

To find the exact value of \( \cos(-60^\circ) \), first recognize that a negative angle means a clockwise rotation. Therefore, \( -60^\circ \) refers to a 60-degree angle in the clockwise direction.

Step 2: Find Reference Angle

The reference angle for \( -60^\circ \) is \( 60^\circ \) itself because reference angles are always positive and less than \( 90^\circ \).

Step 3: Determine Cosine of Reference Angle

The reference angle is \( 60^\circ \). In a standard unit circle, \( \cos(60^\circ) = \frac{1}{2} \).

Step 4: Consider the Quadrant

Since \( -60^\circ \) lies in the Fourth Quadrant (where cosine is positive), the cosine value remains the same as that of the reference angle. Hence, \( \cos(-60^\circ) = \frac{1}{2} \).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \), is one of the primary trigonometric functions used in mathematics to relate the angles of a triangle to the lengths of its sides. It is especially useful in right-angled triangles, where it represents the ratio of the length of the adjacent side to the hypotenuse.

• In a right triangle, for an angle \( \theta \), \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).
• Cosine is a periodic function with a period of \(360^°\) (or \(2\pi\) for radians), which means it repeats its values every complete circle.
• Another important aspect of cosine is that its value ranges between -1 and 1, inclusive.

For negative angles, cosine values are determined the same way as positive angles, but based on the direction of rotation from the x-axis. Hence, \( \cos(-\theta) = \cos(\theta) \) when considering the symmetry of the cosine function about the y-axis.

Unit Circle

The unit circle is a valuable tool for understanding and using trigonometric functions like cosine. It is a circle with a radius of one unit centered at the origin of a coordinate plane.

• Any point on the circumference of the unit circle can be represented by \((\cos(\theta), \sin(\theta))\). This representation makes it easier to find the cosine and sine of various angles.
• The unit circle divides the plane into four quadrants, each affecting the signs of sine and cosine differently. For example, in the first and fourth quadrants, the cosine values are positive, while in the second and third they are negative.
• Using the unit circle, we can visualize angles in standard position starting from the positive x-axis, moving counterclockwise.

When determining the cosine of angles like \(-60^°\), the unit circle helps identify in which quadrant this angle lies and how its cosine value should be considered.

Reference Angle

The reference angle is the positive acute angle formed between the terminal side of a given angle and the closest x-axis. It plays a crucial role in simplifying the calculation of trigonometric functions.

• The reference angle is always between \(0^°\) and \(90^°\), thus always a positive, acute angle.
• For angles in standard position, the reference angle can be calculated by finding the difference between the given angle and the nearest multiple of \(180^°\).
• In solving \(\cos(-60^°)\), the reference angle is \(60^°\), since it is already an acute angle and lies 60 degrees from the x-axis even when rotating clockwise.

This concept simplifies the process of determining trigonometric values because the cosine (and sine) can directly use these reference angles along with appropriate signs from the quadrant.

Angle in Standard Position

An angle is said to be in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis.

• These angles are typically measured from the positive x-axis, proceeding counterclockwise.
• However, when an angle is represented in a clockwise direction, it is considered negative.
• For example, \(-60^°\) would be in the fourth quadrant when represented in standard position because it measures 60 degrees clockwise from the positive x-axis.

Understanding angles in standard position is fundamental for correctly applying trigonometric functions, as the unit circle relies on this positioning to determine the correct quadrant and corresponding sine or cosine values.