Problem 17 - Algebra 2

Question

Use a half-angle identity to find the exact value of each expression. $$ \cos 90^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 90^{\circ} \) is 0.

Step 1: Identify the angle

Recognize that the angle being asked is 90 degrees which is a basic angle on the unit circle. In this special case, the cosine value can be directly determined.

Step 2: Determine the cosine value

By recalling the cosine values for basic angles on the unit circle, we know that at 90 degrees, the cosine value is 0. The cosine function reaches its zeroes at 90 and 270 degrees.

Step 3: Final result

Therefore, the exact value of \( \cos 90^{\circ} = 0 \).

Key Concepts

Half-Angle Identity

In trigonometry, the half-angle identities are useful tools for simplifying expressions and solving equations involving trigonometric functions. These identities help us find the sine, cosine, or tangent of half an angle, given the original angle's trigonometric functions.For cosine, the half-angle identity is given by:\[\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}\]This formula enables us to find the cosine of half of any angle \( \theta \) by using the cosine of the full angle.

• The plus or minus sign depends on the angle's quadrant.
• For angles between 0° and 180°, the cosine of their half-angle is positive.
• For angles between 180° and 360°, it is negative.

Although the half-angle identity is primarily used when the exact value at half of a known angle needs to be calculated, it's good to recognize this identity’s application as a tool to derive fundamental trigonometric values. For instance, calculating \( \cos 45° \) involves using \( 90° \), where cosine is zero. In our particular exercise, we confirmed \( \cos 90° \) was 0 through direct observation rather than formal use of the half-angle formula as the value was trivial.

Unit Circle

The unit circle is a fundamental concept in trigonometry, serving as a visual and computational tool that simplifies the understanding of trigonometric functions.

• It is a circle with a radius of 1, centered at the origin of a coordinate plane.
• The horizontal axis represents the cosine values, whereas the vertical axis represents sine values for a given angle.

Angles in trigonometry are often measured from the positive x-axis, moving counterclockwise. In the context of this exercise, the unit circle helps us understand why the cosine of 90 degrees equates to zero. At 90°: - The point on the unit circle is directly above the origin. - The x-coordinate, which represents the cosine value, is 0, as the point lies directly on the y-axis. Understanding the unit circle allows learners to visualize and internalize why certain trigonometric values exist at specific angles. For angles like 90° and 270°, we can see they lie on the axes where either sine or cosine is zero, effectively removing the need for lengthy calculations.

Cosine Function

The cosine function is one of the primary trigonometric functions, offering vital insights into angle measurements and their computations.

• It maps an angle to the x-coordinate of the corresponding point on the unit circle.
• Cosine is even, meaning \( \cos(-\theta) = \cos(\theta) \), which implies symmetry around the y-axis.

In practical terms, for a given angle \( \theta \), the cosine function can be used to determine how far a point is horizontally from the origin. In the context of this exercise, understanding the cosine function directly provides the value for 90° without complex operations:- At 90°, the cosine function output is 0 because the point on the unit circle is directly above the origin on the y-axis.This makes the cosine function particularly useful for solving various problems, such as those involving harmonic motion, waves, and circular movements, where angles can reflect wave phases or rotations.