Problem 10 - Precalculus

Question

Use degree measure for your answers. In parts (c) and (d), use a calculator and round the results to one decimal place. (a) In \(\triangle A B C, \sin B=\sqrt{3} / 2 .\) What are the possible values for \(\angle B ?\) (b) In \(\triangle D E F, \cos E=-\sqrt{3} / 2 .\) What are the possible values for \(\angle E ?\) (c) In \(\triangle G H I, \sin H=2 / 9 .\) What are the possible values for \(\angle H ?\) (d) In \(\triangle J K L, \cos K=2 / 3 .\) What are the possible values

Step-by-Step Solution

Answer

(a) 60° or 120°; (b) 150° or 210°; (c) 12.9° or 167.1°; (d) 48.2° or 131.8°.

Step 1: Understanding the Problem for Part (a)

We need to find the angles in a triangle where the sine of angle B is given as \( \sin B = \frac{\sqrt{3}}{2} \). This sine value is characteristic of specific angles on the unit circle.

Step 2: Finding Possible Values for \( \angle B \)

The sine of an angle is \( \frac{\sqrt{3}}{2} \) at angles of \( 60^\circ \) and \( 120^\circ \) in the unit circle. Thus, possible values for \( \angle B \) are \( 60^\circ \) and \( 120^\circ \).

Step 3: Understanding the Problem for Part (b)

We need to determine the angles where the cosine is \( \cos E = -\frac{\sqrt{3}}{2} \). Since cosine corresponds to the x-coordinate on the unit circle, negative values occur in the second and third quadrants.

Step 4: Finding Possible Values for \( \angle E \)

Cosine is \( -\frac{\sqrt{3}}{2} \) at angles of \( 150^\circ \) and \( 210^\circ \) on the unit circle. Therefore, possible values for \( \angle E \) are \( 150^\circ \) and \( 210^\circ \).

Step 5: Understanding the Problem for Part (c)

In this part, \( \sin H = \frac{2}{9} \). To find \( \angle H \), we need to use a calculator to find the angle whose sine is \( \frac{2}{9} \).

Step 6: Calculating \( \angle H \)

Using a calculator, apply the inverse sine function: \( \angle H = \sin^{-1}\left(\frac{2}{9}\right) \). This gives \( \angle H \approx 12.9^\circ \). The second quadrant angle, using \( 180^\circ - \angle H \), is approximately \( 167.1^\circ \).

Step 7: Understanding the Problem for Part (d)

For part (d), \( \cos K = \frac{2}{3} \). To find \( \angle K \), use the inverse cosine function.

Step 8: Calculating \( \angle K \)

Use a calculator to find \( \angle K = \cos^{-1}\left(\frac{2}{3}\right) \). This results in \( \angle K \approx 48.2^\circ \). The second quadrant angle, using \( 360^\circ - \angle K \), is possible, but since we are dealing with a triangle, only \( 180^\circ - 48.2^\circ = 131.8^\circ \) is valid.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand angles and their corresponding trigonometric values. It is a circle with a radius of one centered at the origin of a coordinate system. The circle allows us to define trigonometric functions for any angle, measured in degrees or radians, by assigning points on the circle based on these angles.
For sine and cosine, which often appear in trigonometry problems, the unit circle provides the sine as the y-coordinate and the cosine as the x-coordinate of points located on the circle.

• Angles like 0°, 90°, 180°, and 270° pinpoint important locations on the unit circle.
• Specific sine and cosine values, such as \(\sin 30^\circ = \frac{1}{2}\) or \(\cos 60^\circ = \frac{1}{2}\), are defined by specific points on the circle.

The unit circle extends these basic principles to encompass all angles, assisting in solving trigonometric problems, like determining angles with specific sine or cosine values.

Triangle Angles

When working with triangle angles, especially in trigonometry problems, we often look to find specific angle measures using trigonometric functions. Angles in triangles can take on values based on their trigonometric sine or cosine expressions.
In triangles:

• The sum of angles always adds up to 180 degrees.
• Acute angles are less than 90 degrees, while obtuse angles are greater than 90 degrees but less than 180 degrees.

To find possible triangle angles when given a sine or cosine value, one uses the properties of these functions from the unit circle.
For example:

• When \(\sin \angle B = \frac{\sqrt{3}}{2}\), the possible angles could be \(60^\circ\) and \(120^\circ\).
• When \(\cos \angle E = -\frac{\sqrt{3}}{2}\), angles could be \(150^\circ\) and \(210^\circ\).

Understanding these concepts is crucial for solving geometric problems and connecting angle measures with trigonometric ratios.

Inverse Trigonometric Functions

Inverse trigonometric functions are mathematical tools that help us determine an angle when given a trigonometric value. These functions include inverse sine \(\sin^{-1}(x)\), inverse cosine \(\cos^{-1}(x)\), and inverse tangent \(\tan^{-1}(x)\), which are also known as arcsin, arccos, and arctan, respectively.
These functions are particularly useful in trigonometry and geometry when:

• We need to solve for an angle corresponding to a non-standard trigonometric ratio.
• A calculator can be utilized to directly compute these angles, often resulting in decimal values rounded to the nearest degree.

An example from the original exercise:

• If \(\sin H = \frac{2}{9}\), we use \(\angle H = \sin^{-1}\left(\frac{2}{9}\right)\), calculated as approximately \(12.9^\circ\).
• For \(\cos K = \frac{2}{3}\), the angle \(\angle K = \cos^{-1}\left(\frac{2}{3}\right)\) results in approximately \(48.2^\circ\).

Understanding how to apply inverse trigonometric functions is essential for finding precise angle measures.

Degree Measure

Degree measure is a way of expressing the size of an angle. In trigonometry, it is the standard unit used to describe angles, often preferred for its simplicity and direct application in geometric calculations.
The full circle is divided into 360 degrees, allowing precise angle measurements from 0 to 360 degrees. Here are a few key points about degree measure:

• Common references include 90 degrees for a right angle, 180 degrees for a straight line.
• Angles can be acute, right, obtuse, or reflex, depending on their measures.

In the context of finding possible angle values using trigonometric functions, degree measure allows us to translate the theoretical trigonometric values from the unit circle into practical, usable angles. For example:

• For the original exercise, finding angles like \(60^\circ\), \(120^\circ\), and others provides specific degree measures of possible triangle angles.
• Degree measure facilitates easy computation and understanding of angles.

Using degree measure is vital in connecting mathematical trigonometric concepts with real-world applications in geometry and beyond.