Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-23,\) use a calculator to find each value of \(\theta\) to the nearest degree. $$ \theta=\arccos (-0.6) $$

Step-by-Step Solution

Answer

The angle \( \theta = \arccos(-0.6) \) is approximately 126 degrees.

Step 1: Understand the Problem

We need to find the angle \( \theta \) for which the cosine value is \(-0.6\). This angle is given by the inverse cosine function, \( \arccos(-0.6) \).

Step 2: Use a Calculator

Use a scientific calculator to compute the inverse cosine of \(-0.6\). Ensure your calculator is set to degree mode before calculating. Input \( \arccos(-0.6) \) into the calculator.

Step 3: Obtain the Degree Value

After calculating \( \arccos(-0.6) \) using the calculator, you should get a degree value. This value is the angle \( \theta \), whose cosine is approximately \(-0.6\).

Step 4: Round to the Nearest Degree

Round the result of \( \theta \) from the calculator to the nearest whole number (degree). If the decimal is 0.5 or greater, round up; otherwise, round down.

Key Concepts

Angle Calculation

When we talk about angle calculation using the inverse cosine function, we're on a quest to determine an angle whose cosine equals a specific value. In this context, we're dealing with the cosine value of -0.6. The method to find this unknown angle, denoted as \(\theta\), involves the inverse cosine, often symbolized as \(\arccos\).

In essence, \(\arccos\) gifts us with an angle when we start with a cosine value. It answers the question: "For a cosine value of -0.6, what is the corresponding angle?" Keep in mind, the range of possible answers for \(\theta\) is between 0 to 180 degrees because the cosine function varies these values within this interval on a unit circle.

Thus, computing \(\arccos(-0.6)\) situates \(\theta\) precisely on a circle where the x-coordinate is -0.6, along the circle's circumference, representing a unique angle in degrees.

Degree Mode

Calculators are nifty tools, but they require proper setting adjustments to deliver accurate results. While calculating angles like \(\theta\), ensuring your calculator is in degree mode is crucial. Degree mode tells your calculator to interpret angles in degrees rather than radians or other units.

Before starting your calculations, verify your calculator's current angle settings. Most scientific calculators have an easy way to switch modes; look for a button labeled "MODE" or check the screen's indicators. If your calculator accidentally computes in radian mode, the results will deviate from the expected degree values and could lead to confusion.

By consistently using degree mode for problems requiring degrees, like our exercise of finding \(\arccos(-0.6)\), we maintain the integrity of our mathematical exploration. This mode is crucial for problems arising in trigonometry where the output needs to align with common angle interpretations employed globally.

Using Scientific Calculator

Using a scientific calculator is straightforward, but understanding its functionality maximizes your efficiency. When tasked with finding \(\theta\) using \(\arccos(-0.6)\), a scientific calculator becomes indispensable.

First, make sure the calculator is in degree mode. Then, clear any previous data by pressing "C" or "AC". Next, identify the \(\arccos\) function button, which might be labeled "2nd" or "shift" along with "cos" to access the inverse function.

Enter -0.6 into the calculator, and press the corresponding \(\arccos\) button. The display should provide a numerical result representing the angle \(\theta\) in degrees.

• Check your calculator's manual if you're unsure about accessing functions.
• Practice entering various values to familiarize yourself with the features.

Finally, after obtaining the result, round it to the nearest whole number to complete your task accurately. Always ensure precision by reviewing your settings before calculation to achieve reliable results in your trigonometric computations.