Problem 15 - Algebra and Trigonometry

Question

13-18 $$ Sketch each triangle, and then solve the triangle using the Law of Sines. $$ \angle A=30^{\circ}, \quad \angle C=65^{\circ}, \quad b=10 $$

Step-by-Step Solution

Answer

\( a \approx 5.025, c \approx 9.1 \).

Step 1: Sketch the Triangle

Draw a triangle labeled \( \triangle ABC \) with vertices labeled \( A, B, \) and \( C \). Mark \( \angle A = 30^{\circ} \) and \( \angle C = 65^{\circ} \). Choose side \( b \) opposite \( \angle B \) with a length of 10 units.

Step 2: Calculate \( \angle B \)

Use the angle sum property of triangles to find \( \angle B \): \( \angle B = 180^{\circ} - 30^{\circ} - 65^{\circ} = 85^{\circ} \).

Step 3: Use the Law of Sines

Apply the Law of Sines to find the unknown sides. The law is given by \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

Step 4: Solve for Side \( a \)

Use \( \frac{a}{\sin 30^{\circ}} = \frac{10}{\sin 85^{\circ}} \). Cross-multiply to get \( a \cdot \sin 85^{\circ} = 10 \cdot \sin 30^{\circ} \). Since \( \sin 30^{\circ} = 0.5 \) and \( \sin 85^{\circ} \approx 0.996 \), solve for \( a \): \[ a = \frac{10 \cdot 0.5}{0.996} \approx 5.025 \].

Step 5: Solve for Side \( c \)

Use \( \frac{c}{\sin 65^{\circ}} = \frac{10}{\sin 85^{\circ}} \). Cross-multiply to get \( c \cdot \sin 85^{\circ} = 10 \cdot \sin 65^{\circ} \). Calculate \( \sin 65^{\circ} \approx 0.906 \), then solve for \( c \):\[ c = \frac{10 \cdot 0.906}{0.996} \approx 9.1 \].

Step 6: Review the Calculations

Verify that the calculated sides are consistent with the initial triangle sketch and angle calculations. The sides should satisfy the Law of Sines and correspond appropriately to their opposite angles.

Key Concepts

Solving Triangles Using Different Angles

Solving a triangle means determining all its unknown sides and angles. This is done by finding out the values based on given information like angles and side lengths. In our example triangle, we know two angles: \( \angle A = 30^{\circ} \) and \( \angle C = 65^{\circ} \), plus the length of one side, which is \( b = 10 \).

To solve the triangle, you first sketch it, labeling each part clearly. This helps visualize the relationships between the angles and sides. Next, you figure out any missing angles using known properties, such as the angle sum property. This forms the foundation for calculating the missing side lengths using further trigonometric principles.

Understanding the Angle Sum Property

The angle sum property of triangles is an essential concept used to solve triangles easily. This property states that the interior angles of a triangle always add up to \( 180^{\circ} \).

When we know two angles, \( \angle A = 30^{\circ} \) and \( \angle C = 65^{\circ} \), we can find \( \angle B \) by subtracting their sum from \( 180^{\circ} \):

• \( \angle B = 180^{\circ} - 30^{\circ} - 65^{\circ} = 85^{\circ} \)

This simple property allows us to complete the set of angles, providing all necessary information to tackle side length calculations. It's a straightforward yet powerful technique necessary for solving any generic triangle configuration.

Trigonometric Methods and Law of Sines

The Law of Sines is particularly useful for finding missing side lengths in a triangle when angles and one side length are known. This law states that the ratio of a side's length to the sine of its opposite angle is the same for all sides of a triangle.

In our triangle:

• \( \frac{a}{\sin 30^{\circ}} = \frac{b}{\sin 85^{\circ}} = \frac{c}{\sin 65^{\circ}} \)

We used this principle to solve for unknown sides \( a \) and \( c \). Applying the Law of Sines involves plugging angles and sides into these ratios and solving for the unknown, making use of the known values like \( \sin 30^{\circ} = 0.5 \). This technique opens up straightforward ways to fully identify any triangle's dimensions.

Calculating Triangle Sides

One of the final steps in triangle solving is calculating the unknown side lengths using angles and the Law of Sines. Once all angles are determined, side calculations become simple:

To find side \( a \):

• Cross-multiply \( \frac{a}{\sin 30^{\circ}} = \frac{10}{\sin 85^{\circ}} \)
• Solve: \( a = \frac{10 \times 0.5}{0.996} \approx 5.025 \)

To find side \( c \):

• Cross-multiply \( \frac{c}{\sin 65^{\circ}} = \frac{10}{\sin 85^{\circ}} \)
• Solve: \( c = \frac{10 \times 0.906}{0.996} \approx 9.1 \)

This method ensures accuracy, allowing verification by checking if calculated sides align with Law of Sines calculations.