Problem 13 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle R S T, t=15.7, s=15.7, \mathrm{m} \angle R=98\)

Step-by-Step Solution

Answer

The area of \( \triangle RST \) is approximately 121.5 square units.

Step 1: Identify the Known Values

We are given a triangle \( \triangle RST \) with sides \( t = 15.7 \), \( s = 15.7 \) and angle \( m \angle R = 98^\circ \). We need to find the area of this triangle.

Step 2: Apply the Formula for Area of a Triangle with SAS

Since two sides and the included angle are known, we use the formula:\[ \text{Area} = \frac{1}{2} \, ab \, \sin C \]For \( \triangle RST \), substitute \( a = s = 15.7 \), \( b = t = 15.7 \), and \( C = \angle R = 98^\circ \).

Step 3: Calculate the Sine of the Angle

Calculate \( \sin 98^\circ \) using a calculator:\[ \sin 98^\circ \approx 0.9867 \]

Step 4: Substitute and Solve

Substitute back into the area formula:\[\text{Area} = \frac{1}{2} \times 15.7 \times 15.7 \times 0.9867 \]Calculate the area:\[ \text{Area} \approx \frac{1}{2} \times 15.7 \times 15.7 \times 0.9867 \approx 121.5 \]

Step 5: Round to the Nearest Tenth

The calculated area is approximately 121.5. When rounded to the nearest tenth, the area of \( \triangle RST \) is 121.5.

Key Concepts

Sine Rule

The Sine Rule is a powerful trigonometric tool used in dealing with non-right angled triangles. It's especially helpful when you know:

• Two angles and one side (AAS or ASA condition)
• Two sides and a non-included angle (SSA condition)

The rule states that the ratio of each side to the sine of its corresponding opposite angle is the same for all three sides of the triangle. It can be expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Remember, using the Sine Rule can help find unknown sides or angles in a triangle. Just make sure the triangle's conditions fit one of the two scenarios for applying it.

SAS Formula

The SAS (Side-Angle-Side) formula is ideal for finding the area of a triangle when you know:

• Two sides
• The angle between them

This specific combination allows you to use the formula:\[ \text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin C \]Here, \(a\) and \(b\) are the side lengths, and \(C\) is the included angle. It's a very efficient method because it reduces the complexity of calculations by using known values directly in a straightforward manner. By knowing these values, you can quickly compute the triangle's area without needing additional information.

Trigonometry

Trigonometry involves the study of the relationships between the sides and angles of triangles. This branch of mathematics is crucial in many fields, including physics, engineering, and architecture. At its core, trigonometry revolves around three primary functions:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

Each function relates an angle with specific ratios of side lengths in right-angled triangles. However, these functions also extend to non-right triangles through laws like the Sine Rule and Cosine Rule. Mastering these concepts aids in solving many real-world problems involving triangles.

Angle Measurement

Understanding how angles are measured is key in using trigonometric principles effectively. Angles are usually measured in degrees or radians.

• Degrees: One full circle is 360 degrees.
• Radians: One full circle is \(2 \pi\) radians.

In trigonometry, angles often define the relationship between the triangle's sides. Knowing the angle helps you apply numerous trigonometric formulas, such as the SAS formula or Sine Rule. Calculators usually have functions to switch between these units, allowing for flexible calculation in different scenarios. Moreover, proper decimal or fractional notation brings precision to computations, especially important when calculating functions like sine or cosine.