Problem 4 - Contemporary Precalculus

Question

Directions: Standard notation for triangle \(A B C\) is used throughout. Use a calculator and round off your answers to one decimal place at the end of the computation. Solve the triangle ABC under the given conditions. $$C=52.5^{\circ}, a=6.5, b=9$$

Step-by-Step Solution

Answer

Question: Find the angles and sides of triangle ABC, given that angle C is \(52.5^{\circ}\), side a is 6.5, and side b is 9. Answer: The triangle has angles A = \(47.3^{\circ}\), B = \(80.2^{\circ}\), C = \(52.5^{\circ}\), and sides a = 6.5, b = 9, and c = 7.3.

Step 1: Apply Law of Sines to find angle A

We can use the Law of Sines to find angle A: $$\frac{a}{\sin(A)} = \frac{b}{\sin(C)}$$ Plug in the known values: $$\frac{6.5}{\sin(A)} = \frac{9}{\sin(52.5^{\circ})}$$ Now, solve for \(\sin(A)\): $$\sin(A) = \frac{6.5 \sin(52.5^{\circ})}{9}$$ Calculate \(\sin(A)\): $$\sin(A) = 0.7356$$ Next, find angle A by taking the inverse sine of the result: $$A = \arcsin(0.7356)$$ $$A = 47.3^{\circ}$$

Step 2: Find angle B using angle sum property

In a triangle, the sum of all angles is \(180^{\circ}\). Now we can find angle B: $$A + B + C = 180^{\circ}$$ Substitute the values of A and C: $$47.3^{\circ} + B + 52.5^{\circ} = 180^{\circ}$$ Solve for B: $$B = 180^{\circ} - 47.3^{\circ} - 52.5^{\circ}$$ $$B = 80.2^{\circ}$$

Step 3: Apply Law of Sines again to find side c

Now, using the Law of Sines, find side c: $$\frac{c}{\sin(C)} = \frac{a}{\sin(A)}$$ Plug in the known values: $$\frac{c}{\sin(52.5^{\circ})} = \frac{6.5}{\sin(47.3^{\circ})}$$ Now, solve for c: $$c = \sin(52.5^{\circ}) \cdot \frac{6.5}{\sin(47.3^{\circ})}$$ Calculate side c: $$c = 7.3$$ Now, we have completely solved the triangle ABC. The triangle has angles A = \(47.3^{\circ}\), B = \(80.2^{\circ}\), C = \(52.5^{\circ}\) and sides a = 6.5, b = 9, and c = 7.3.

Key Concepts

Angle Calculation in Triangles

To understand triangles, calculating angles is essential. In a triangle, each angle can influence the others. Knowing at least one side length and one angle, such as in triangle ABC, you can calculate other unknown angles using trigonometric concepts. For example, applying the Law of Sines simplifies this process greatly. The step-by-step solution uses the proportion \[ \frac{a}{\sin(A)} = \frac{b}{\sin(C)} \]Substitute known values to find angle A:

• Use side a (6.5) and angle C (52.5°).
• Calculate \( \sin(A) \) as \( 0.7356 \).
• Use the inverse sin function to find that \( A = 47.3° \).

Trust your calculations to get accurate angles. This accuracy is key to successfully solving triangles.

Triangle Angle Sum Property

A fundamental property of triangles is that the sum of their angles is always 180°. This principle is incredibly useful. It allows you to find an unknown angle if the other two are known.
In our triangle ABC, we utilize this property by knowing angles A and C:

• Given \( A = 47.3° \) and \( C = 52.5° \)
• Apply the triangle angle sum property: \( A + B + C = 180° \)
• Calculate angle B: \( B = 180° - 47.3° - 52.5° \)
• This gives \( B = 80.2° \)

By understanding and applying this property, solving triangles becomes easier and more straightforward, even when missing an angle.

Solving Triangles Using Trigonometry

Trigonometry is a powerful tool for solving triangles, allowing us to find angles and side lengths. When side and angle details are given, the Law of Sines is particularly valuable:

• The proportion \( \frac{c}{\sin(C)} = \frac{a}{\sin(A)} \) helps find unknown side c.
• Substitute known values: sides a, c, and angle C.
• The calculation uses the sine of known angles to find the third side, ensuring the triangle is accurate: \( c = 7.3 \).

The combination of angle calculation, angle sum property, and trigonometry skills is essential for solving triangles fully. Mastery of these concepts ensures accurate solutions and deeper understanding of geometric relationships.