Problem 6 - Precalculus with Calculus Previews

Question

Use the Law of Sines to solve the triangle. $$ \alpha=80^{\circ}, \beta=20^{\circ}, b=7 $$

Step-by-Step Solution

Answer

Angle \(\gamma = 80^\circ\), side \(a \approx 20.108\), side \(c \approx 20.108\).

Step 1: Calculate the third angle

The sum of the angles in a triangle is always \(180^\circ\). We know \(\alpha = 80^\circ\) and \(\beta = 20^\circ\). Therefore, the third angle, \(\gamma\), can be calculated using the formula: \( \gamma = 180^\circ - \alpha - \beta \). Substituting the known values, we get \( \gamma = 180^\circ - 80^\circ - 20^\circ = 80^\circ \).

Step 2: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \). We are given \( b = 7 \), \( \alpha = 80^\circ \), and \( \beta = 20^\circ \). Therefore, \( \frac{a}{\sin(80^\circ)} = \frac{7}{\sin(20^\circ)} \). We can solve for \( a \) by rearranging this to \( a = \frac{7 \cdot \sin(80^\circ)}{\sin(20^\circ)} \).

Step 3: Calculate side a

Using the equation from Step 2, substitute the sine values: \( a = \frac{7 \cdot \sin(80^\circ)}{\sin(20^\circ)} \). Calculate the sine values and solve for \( a \): \( a \approx \frac{7 \cdot 0.9848}{0.342} \approx 20.108 \).

Step 4: Calculate side c using Law of Sines

Using \( \frac{c}{\sin(80^\circ)} = \frac{7}{\sin(20^\circ)} \), solve for \( c \): \( c = \frac{7 \cdot \sin(80^\circ)}{\sin(20^\circ)} \). Since \( \sin(80^\circ) = 0.9848 \) and \( \sin(20^\circ) = 0.342 \), calculate \( c \): \( c \approx \frac{7 \cdot 0.9848}{0.342} \approx 20.108 \).

Key Concepts

Triangle Solving

To tackle problems involving triangles, understanding how to solve a triangle is key. Solving a triangle involves finding unknown values, such as side lengths or angles, using known values and mathematical principles. For a basic triangle, we often have a combination of given angles and at least one side. To solve the triangle, you need to:

• Determine all angles.
• Calculate all side lengths.

This process typically employs laws like the Law of Sines or Law of Cosines. The Law of Sines is especially useful when you have a situation with two known angles and a side, commonly referred to as AAS (angle-angle-side) or ASA (angle-side-angle) conditions. In these cases, once you know two angles, the third is simply found by subtracting the sum of the known angles from 180°, because a triangle's angles always add up to 180°. With all angles known, solving for the sides becomes straightforward with the Law of Sines.

Angle Calculation

Calculating angles in a triangle is often the first step in solving the triangle. In our exercise, we are given two angles: \( \alpha = 80^{\circ} \) and \( \beta = 20^{\circ} \). Using the fact that the sum of angles in a triangle is always \( 180^{\circ} \), we can easily find the third angle \( \gamma \).
\[ \gamma = 180^{\circ} - \alpha - \beta \] By substituting the values, \( \gamma = 180^{\circ} - 80^{\circ} - 20^{\circ} = 80^{\circ} \). It's crucial to ensure accuracy, as angle calculation sets the stage for using sine laws effectively. Ensuring these steps are thorough and clear helps prevent mistakes later in the problem-solving process.

Trigonometry

Trigonometry acts as the bridge for solving triangles by establishing relationships between angles and sides. The Law of Sines is a trigonometric formula that states:
\[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] This law helps find unknown sides or angles when given part of the triangle's dimensions. For instance, if you have one side and its opposite angle, as is the case in our example with side \( b = 7 \) and angle \( \beta = 20^{\circ} \), you can determine the other side \( a \) using the corresponding angle \( \alpha = 80^{\circ} \).

Calculations such as:
\[ a = \frac{7 \cdot \sin(80^{\circ})}{\sin(20^{\circ})} \] provide the desired side length, where known sine values \( \sin(80^{\circ}) = 0.9848 \) and \( \sin(20^{\circ}) = 0.342 \) allow us to compute \( a \approx 20.108 \). Knowledge of trigonometry streamlines solving these equations, making it essential for comprehending the spatial relationships in geometry.