Problem 81 - Algebra and Trigonometry

Question

Exercises 81–83 will help you prepare for the material covered in the next section. Find the obtuse angle \(B,\) rounded to the nearest degree, satisfying $$ \cos B=\frac{6^{2}+4^{2}-9^{2}}{2 \cdot 6 \cdot 4} $$

Step-by-Step Solution

Answer

The obtuse angle B satisfying the equation \( cos B = \frac{6^{2} + 4^{2} - 9^{2}}{2 \cdot 6 \cdot 4} \) is 141° (rounded to the nearest degree).

Step 1: Identify Given Values

Identify the given values in the problem. Here we have a=6, c=4, and b=9.

Step 2: Apply the Cosine Rule

The Law of Cosines states \(cos B = \frac{a^{2} + c^{2} - b^{2}}{2 a c}\). Substitute the given values into the equation to find the cosine of B. So, \(cos B = \frac{(6)^{2} + (4)^{2} - (9)^{2}}{2 \cdot 6 \cdot 4}\). Compute the values.

Step 3: Calculate Inverse Cosine

Use the inverse cosine function, written as \(cos^{-1}\), to determine the measure of angle B. Keep in mind that the resulting angle will be in radians.

Step 4: Convert to Degrees

If needed, convert B from radians to degrees by multiplying by \(\frac{180}{\pi}\). Round off the result to the nearest degree as instructed.

Step 5: Determine Obtuseness

As per the definition, an obtuse angle is an angle between 90° and 180°. So, check if your answer lies in this range and adjust as necessary.

Key Concepts

Obtuse Angles

An obtuse angle is any angle greater than 90° but less than 180°. These angles are found between the two directions of a straight line. Unlike acute angles, which are small and sharp, obtuse angles are larger and more open.
To find an obtuse angle in a triangle, particularly using the Law of Cosines, you often look for the angle opposite the longest side. In this exercise, if the calculated angle B is greater than 90°, you have confirmed it's obtuse.
Recognizing obtuse angles in problems illustrates how understanding angle properties can help verify solutions in geometric problems.

Inverse Cosine Function

The inverse cosine function, denoted as \( ext{cos}^{-1}\), allows you to find an angle when you know its cosine value. It helps retrieve the unique angle whose cosine is a given number, usually over the interval \([0, \, ext{arccos}]\).
When working with the inverse cosine function, keep in mind that the range is from 0 to \( \pi \) radians (or 0° to 180°). This is crucial for identifying whether the resulting angle is acute, right, or obtuse.

• \( \text{cos}^{-1}(x) \) yields an angle when \(-1 \leq x \leq 1\).
• Applications are often found in triangle problems and curve computations.

Angle Conversion from Radians to Degrees

Converting angles from radians to degrees is often necessary in trigonometry, as degrees are a more intuitive measure for many people. The formula for conversion is straightforward: simply multiply the angle in radians by \(rac{180}{\pi}\).
For example, if you calculate an angle using \(\text{cos}^{-1}\) and the result is in radians, apply the conversion to express it in degrees. This is essential when problems require the answer to be in degrees, such as in the given exercise.

• Degrees are based on dividing a circle into 360 parts.
• Radians are based on the radius of the circle, more common in higher-level mathematics.

Understanding both radians and degrees, and how to convert between them, is a valuable skill in mathematics.