Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arccos (-1) $$

Step-by-Step Solution

Answer

\( \theta = 180^\circ \) in degrees and \( \pi \) in radians.

Step 1: Understanding the Problem

We need to find the angle \( \theta \) such that \( \theta = \arccos(-1) \). We will calculate \( \theta \) in both degrees and radians.

Step 1: Interpretation of \( \arccos(-1) \)

The function \( \arccos \) refers to the angle whose cosine value is given. For \( \arccos(-1) \), we need to find the angle where the cosine is \(-1\).

Step 2: Calculating \( \theta \) in Degrees

The cosine of \( 180^\circ \) is \(-1\). Therefore, \( \arccos(-1) = 180^\circ \). So, \( \theta = 180^\circ \) in degrees.

Step 3: Converting Degrees to Radians

To convert degrees to radians, we use the conversion formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). For \( \theta = 180^\circ \), \( \theta = 180 \times \frac{\pi}{180} = \pi \) radians.

Step 5: Conclusion

Thus, the value of \( \theta \) is \( 180^\circ \) in degrees and \( \pi \) in radians.

Key Concepts

Angles in Degrees

Degrees are a measure for angles that are widely used in many fields like navigation, geometry, and trigonometry. One complete circle is divided into 360 equal parts, each called a degree, denoted by the symbol 0^.

• Each quarter of a circle, or one right angle, measures 90 degrees.
• Half of a circle, known as a straight angle, is 180 degrees.
• Three quarters, known as a reflex angle, is 270 degrees.

In trigonometry, the cosine of any angle can be found using the unit circle. The angle of 180 degrees has a cosine value of -1. This is because on the unit circle, a 180-degree rotation aligns with the point (-1, 0) on the x-axis.
When dealing with the \(\arccos\) function, we find the angle whose cosine is -1, leading us to the solution that \(\arccos(-1) = 180^\circ\).

Angles in Radians

Radians provide another way to measure angles, and they are often preferred in mathematics because they result in simpler formulas for many calculations. Radians relate the arc length to the radius in a circle.

• A full circle in radians is given by the circumference divided by the radius, which is 2\(\pi\) radians.
• Half a circle, or a straight angle, is represented as \(\pi\) radians.
• One quarter of a circle, or a right angle, is \(\frac{\pi}{2}\) radians.

To convert degrees into radians, use the conversion formula:\[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]For example, if you convert 180 degrees into radians, you multiply 180 by \(\frac{\pi}{180}\), resulting in \(\pi\) radians. Therefore, \(\arccos(-1)\) in radians equals \(\pi\).

Unit Circle

The unit circle is a powerful tool in trigonometry often used to define the trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle allows us to find trigonometric values for different angles.

• Points on the unit circle are written as \((x, y)\), where \(x\) represents \(\cos(\theta)\) and \(y\) represents \(\sin(\theta)\).
• For an angle of 180 degrees, or \(\pi\) radians, the coordinates are (-1, 0).
• On the unit circle, \(\arccos(-1)\) corresponds to an angle that brings us to the point (-1, 0)\1, illustrating why \(\theta = 180^\circ\) or \(\pi\) radians.

The power of the unit circle comes from its ability to provide geometric insight and a straightforward way to calculate trigonometrical values across the full circle, for both positive and negative angles. It serves as a graphical representation of angle measurement, linking both radians and the unit angle.