Problem 11 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s), if possible, using any appropriate technique. $$ a=18, \alpha=63^{\circ}, b=20 $$

Step-by-Step Solution

Answer

The remaining side is \(c = 11.52\), and the angles are \(\beta = 82.3^{\circ}\) and \(\gamma = 34.7^{\circ}\).

Step 1: Identify the Known Quantities

We have side \(a = 18\), angle \(\alpha = 63^{\circ}\), and side \(b = 20\). We need to find side \(c\) and angles \(\beta\) and \(\gamma\).

Step 2: Find the Angle \(\beta\) Using the Law of Sines

The Law of Sines states \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \). Substitute the known values: \( \frac{18}{\sin 63^{\circ}} = \frac{20}{\sin \beta} \). Solve for \(\sin \beta\) by cross-multiplying: \( \sin \beta = \frac{20 \sin 63^{\circ}}{18} \). Calculate \(\beta\).

Step 3: Calculate \(\beta\)

Calculate \( \sin \beta = \frac{20 \times 0.8910}{18} = 0.990 \). Calculate \(\beta\) using inverse sine: \( \beta = \arcsin(0.990) = 82.3^{\circ} \).

Step 4: Find Angle \(\gamma\)

Since the sum of the angles in a triangle is \(180^{\circ}\), use \( \gamma = 180^{\circ} - \alpha - \beta = 180^{\circ} - 63^{\circ} - 82.3^{\circ} = 34.7^{\circ} \).

Step 5: Find Side \(c\) Using the Law of Sines

Use \( \frac{c}{\sin \gamma} = \frac{a}{\sin \alpha} \). Substitute known values: \( \frac{c}{\sin 34.7^{\circ}} = \frac{18}{\sin 63^{\circ}} \). Solve for \(c\): \( c = \frac{18 \times 0.5708}{0.8910} = 11.52 \).

Step 6: Calculations Complete

We now have all the sides and angles: \(b = 20\), \(a = 18\), \(c = 11.52\), \(\alpha = 63^{\circ}\), \(\beta = 82.3^{\circ}\), and \(\gamma = 34.7^{\circ}\).

Key Concepts

Triangle Solving

Solving a triangle involves finding the unknown elements, which can be sides or angles, using known measurements. In this exercise, we're given two sides and one angle in a non-right triangle. This scenario is typically solved using the Law of Sines or the Law of Cosines, depending on what information is available.

Here's a brief roadmap to solving a triangle like the one in our exercise:

• Identify the type of triangle: In our case, it's an oblique triangle, which means no angle is a right angle.
• Decide on the method: Since we're given two sides and an angle opposite one of them, the Law of Sines is well-suited here.
• Use trigonometric functions: Sine, cosine, and tangent will be your best friends when dealing with angles and sides in triangles.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables. In the context of triangle solving, the Law of Sines is a golden resource.

The Law of Sines identifies the relationship between the sides and their opposite angles, stating that for a triangle with sides \( a, b, c \) and opposite angles \( \alpha, \beta, \gamma \), it holds that:

• \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \)

This equation helps us find missing sides or angles when we have enough information to set up a correct proportion.

Here’s a quick tip for using trig identities:

• Always ensure your angles are in the right mode (degrees if using degrees or radians for radians)
• Remember, inverse trigonometric functions, like \( \arcsin \), are used to find angles from a given sine value.

Angle Calculation

Angle calculation is crucial for solving triangles, as knowing all angles helps in pinpointing unknown sides and verifying results. In our triangle, after calculating one missing angle \( \beta \), the other, \( \gamma \), can be found easily.

Angle calculation steps:

• The sum of angles in any triangle is always \( 180^{\circ} \). This property allows you to find the third angle if the other two are known.
• In our problem, after calculating \( \beta \) using the Law of Sines, use the formula \( \gamma = 180^{\circ} - \alpha - \beta \) to find \( \gamma \).
• Always double-check your work with calculated angles to ensure they sum up correctly, adding up to \( 180^{\circ} \).

Correctly handling angle calculations is essential to ensure all sides also measure up to your calculated angles.