Problem 24 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos 0\)

Step-by-Step Solution

Answer

The exact value of \(y = \arccos 0\) is \(\frac{\pi}{2}\).

Step 1: Understand the Problem

We need to find the value of \(y = \arccos 0\). This represents the angle whose cosine is 0. Our goal is to find this value in terms of \(\pi\).

Step 2: Recall the Cosine Function

Recall that the cosine of an angle \(\theta\) in the unit circle is the x-coordinate of the point on the circle. We need to determine for which angles the x-coordinate is zero.

Step 3: Identify Angles with Cosine 0

In the unit circle, the cosine value is zero at angles \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\). The \(\arccos\) function, however, only outputs values in the range \([0, \pi]\) (the principal value).

Step 4: Find the Principal Value

The principal value of \(\arccos 0\) in the interval \([0,\pi]\) is \(\frac{\pi}{2}\), where the cosine is 0.

Key Concepts

Understanding Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees that divide a circle into 360 parts, radians relate directly to the circle's circumference.
One full circle around the unit circle is equal to 2\(\pi\) radians, which is the same as 360 degrees. Therefore, knowing that \(\pi\) radians is equivalent to 180 degrees can help you convert between radians and degrees easily.
The use of radians is common in higher mathematics because they provide a natural way to describe angles in terms of the properties of the circle itself. This relationship between circle and angle measurement makes calculations more straightforward in calculus and trigonometry.

Exploring the Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is used to define the trigonometric functions for all angles.

• The equation of the unit circle is \(x^2 + y^2 = 1\).
• Every point on the unit circle corresponds to an angle that originates from the positive x-axis.
• The x-coordinate of any point on the unit circle represents the cosine of the angle.
• The y-coordinate of any point on the unit circle represents the sine of the angle.

Thus, the unit circle helps visualize angles, making it easier to comprehend how trigonometric functions such as cosine and sine work.

Breaking Down the Cosine Function

The cosine function is one of the main trigonometric functions. It calculates the x-coordinate of a point on the unit circle for any given angle. Think of the cosine function as a way to determine how far left or right a point is on the circle as you measure from the center.

• In the unit circle, \(\cos(\theta)\) gives the horizontal distance from the origin.
• The range of the cosine function is from -1 to 1.
• The cosine function is periodic, with a period of \(2\pi\).

When dealing with inverse cosine, or \(\arccos\), it finds an angle whose cosine is a given value, typically within the range of 0 to \(\pi\). This inverse function allows us to translate a known cosine value back into an angle measurement.

Decoding the Principal Value

The principal value is a way to narrow down the range of the inverse trigonometric functions to ensure they output a single, distinctive value. For \(\arccos\), the principal value falls within the interval \([0, \pi]\). This means arc cosine can return only the angle that lies in the upper half of the unit circle.
In this context, when you solve \(y = \arccos 0\), you'll look for the angle in this specific range where the cosine is exactly 0. Both \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) radians correspond to \(\cos(\theta) = 0\), but \(\frac{3\pi}{2}\) is outside the principal range. Consequently, the correct and unique output is \(\frac{\pi}{2}\), ensuring clarity whenever the inverse cosine function is applied.