Problem 22 - Algebra and Trigonometry

Question

Find the area of each triangle. Round answers to two decimal places. $$b=4, \quad c=1, \quad A=120^{\circ}$$

Step-by-Step Solution

Answer

The area is approximately 1.73 square units.

Step 1: Identify the given information

The given values are: side b = 4, side c = 1, and angle A = 120°.

Step 2: Use the formula for area of a triangle with given sides and included angle

The formula to find the area of a triangle when two sides and the included angle are known is: \[ \text{Area} = \frac{1}{2} \times b \times c \times \text{sin}(A) \].

Step 3: Substitute the given values into the formula

Substitute b = 4, c = 1, and A = 120° into the formula: \[ \text{Area} = \frac{1}{2} \times 4 \times 1 \times \text{sin}(120°) \].

Step 4: Calculate sin(120°)

Recall that \( \text{sin}(120°) = \text{sin}(180° - 60°) = \text{sin}(60°) = \frac{\text{√3}}{2} \).

Step 5: Perform the multiplication

Now substitute \( \text{sin}(120°) = \frac{\text{√3}}{2} \) into the formula and perform the calculation: \[ \text{Area} = \frac{1}{2} \times 4 \times 1 \times \frac{\text{√3}}{2} = \frac{4 \text{√3}}{4} = \text{√3} \].

Step 6: Round the answer to two decimal places

To find the area to two decimal places, calculate \( \text{√3} \) which is approximately 1.732. Thus, the area ≈ 1.73.

Key Concepts

triangle area formula

One of the most useful formulas in geometry is the triangle area formula, especially when dealing with non-right triangles. For triangles where you know two sides and the included angle, this formula comes in handy: \[ \text{Area} = \frac{1}{2} \times b \times c \times \text{sin}(A) \] Here, \( b \) and \( c \) are the lengths of two sides of the triangle, and \( A \) is the angle between those sides. This formula uses a mix of basic geometry and trigonometry. This makes it versatile for solving many triangle-related problems. It's different from the basic \( \frac{1}{2} \times \text{base} \times \text{height} \) formula, which only works for right triangles or when you know the height.

trigonometry

Trigonometry deals with the relationships between the angles and sides of triangles. In the context of our area formula, it specifically helps us relate angles to side lengths, and makes use of functions like sine. By applying trigonometric identities and functions such as sine, cosine, and tangent, we can solve complex problems involving non-right triangles. Trigonometry is very important in various fields like engineering, physics, and even computer science. So understanding the basics truly benefits you in the long run.

sine function

The sine function, often written as \( \text{sin} \), is a fundamental concept in trigonometry. It relates a given angle in a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. For example, \( \text{sin}(A) = \frac{\text{opposite}}{\text{hypotenuse}} \). In our triangle area formula, using the sine of the included angle helps determine the proportion of the area relative to the sides and the angle. For example, \( \text{sin}(120^\text{°}) \) translates to \( \text{sin}(180^\text{°} - 60^\text{°}) \). Knowing trigonometric identities like these makes calculating areas easier.

included angle

The included angle in a triangle is the angle formed between two known sides. In our exercise, the included angle is 120°. Recognizing and correctly using the included angle is crucial for applying the triangle area formula accurately. It’s important to remember that the included angle is always the one directly between the sides you’re using in your calculations. For instance, if sides \( b \) and \( c \) are given, then \( A \) needs to be the angle between them for the formula \( \text{Area} = \frac{1}{2} \times b \times c \times \text{sin}(A) \) to be valid.