Problem 51 - Trigonometry

Question

Find \(c\) for triangle \(A B C\) if \(A=43^{\circ}, B=12^{\circ}\), and \(b=25\) centimeters. a. \(98 \mathrm{~cm}\) b. \(34 \mathrm{~cm}\) c. \(120 \mathrm{~cm}\) d. \(82 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The length of side \(c\) is approximately 98 cm (option a).

Step 1: Find Angle C

In triangle ABC, the sum of the angles is always 180°. Therefore, we can find angle C by calculating:\[C = 180° - A - B = 180° - 43° - 12°\]So, angle C is 125°.

Step 2: Use the Law of Sines

To find the length of side \(c\), we use the Law of Sines:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]We're given \(b = 25\) cm and are looking for \(c\). Rearranging the law gives:\[c = \frac{b \sin C}{\sin B}\]

Step 3: Calculate Sine of Angles

Compute the sine of angles B and C:\[\sin 12° = 0.2079\quad\text{(approximately)}\]\[\sin 125° = 0.8192\quad\text{(approximately)}\]

Step 4: Solve for Side c

Substitute the values into our rearranged Law of Sines formula:\[c = \frac{25 \times 0.8192}{0.2079}\]This results in:\[c \approx 98.51 \text{ cm}\]

Step 5: Choose the Closest Option

The calculated length of side \(c\) is approximately 98.51 cm. Compare this with the provided choices to find the closest one, which is 98 cm.

Key Concepts

Triangle Angle Sum

In any triangle, the sum of its internal angles always equals 180 degrees. This is known as the triangle angle sum rule and is a fundamental concept in geometry. It's very useful because once you know two angles in a triangle, you can easily find the third one. In the example with triangle ABC, the angles given are \( A = 43^\circ \) and \( B = 12^\circ \). Using the triangle angle sum rule, we find angle C by subtracting the known angles from 180 degrees:

• Step 1: Calculate \( C = 180^\circ - A - B \)
• Step 2: Substitute the values: \( 180^\circ - 43^\circ - 12^\circ \)
• Step 3: Perform the subtraction: \( C = 125^\circ \)

Once all angles are known, we can proceed to use other trigonometric laws to solve for unknown sides of the triangle.

Sine Function

The sine function is one of the primary trigonometric functions and is essential in solving triangles. In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. However, it is also used in non-right triangles as part of the Law of Sines, allowing us to relate side lengths and angles.

• Sine is written as \( \sin \theta \), where \( \theta \) is an angle.
• For small angles, sine values are less than 1, and they increase as the angle approaches 90 degrees.

Knowing the sine value of an angle can let us determine unknown sides or angles within a triangle when we apply the Law of Sines. In our example:

• The sine of \( B = 12^\circ \) is approximately \( 0.2079 \).
• The sine of \( C = 125^\circ \) is approximately \( 0.8192 \).

These values are crucial to finding the length of side \( c \) using further trigonometric relationships.

Angle Calculation

Calculating angles is a straightforward process if you understand the relationships between the angles and sides of a triangle. Once you have all angles in a triangle, you can use them to calculate unknown sides using the Law of Sines:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]With triangle ABC, we use the values of known angles and sides:

• Given: \( b = 25 \) cm, \( \sin B = 0.2079 \), and \( \sin C = 0.8192 \)
• Rearrange to solve for \( c \):

\[c = \frac{b \cdot \sin C}{\sin B} = \frac{25 \cdot 0.8192}{0.2079}\]By substituting these values, we find \( c \approx 98.51 \) cm. This demonstrates how effectively we can use known angles and their sine values to calculate side lengths. Calculations like these are helpful when solving any triangle, ensuring you have the tools to find unknown measures based on given information.