Matematika SMA - Trigonometri (8) - Trigonometri Luas Segitiga, Rumus Luas Segitiga (A)

Summary

In this educational video, the presenter explains how to calculate the area of a triangle using trigonometry. The video covers three methods: Heron's formula (for when all sides are known), a formula for when two angles and one side are known, and a method for when two sides and the included angle are known. The video includes clear examples demonstrating each approach, ensuring viewers understand how to apply these formulas in real-world problems. The tutorial is ideal for anyone looking to deepen their understanding of trigonometric principles for geometry.

Takeaways

• 😀 Trigonometry is applied to calculate the area of a triangle in various ways.
• 😀 The first method to calculate the area of a triangle uses Heron's formula when all three sides are known.
• 😀 Heron's formula requires calculating the semi-perimeter and applying it with the sides of the triangle to find the area.
• 😀 The second method is used when two angles and one side of the triangle are known, involving the use of sine functions.
• 😀 The third method calculates the area when two sides and the angle between them are known, commonly using the formula: 1/2 * a * b * sin(C).
• 😀 An example using Heron's formula is provided where the sides of the triangle are 6, 8, and 4 units.
• 😀 In another example, the area is found using two angles (60° and 37°) along with the side length of 12 cm.
• 😀 A third example demonstrates calculating the area when two sides (12 cm and 8 cm) and the angle between them (45°) are known.
• 😀 The instructor uses practical examples to demonstrate each formula and guide students through step-by-step solutions.
• 😀 In the final problem, a triangle with a right angle is used, and the goal is to find the length of a segment using known areas and trigonometric relations.

Q & A

What is the basic formula for calculating the area of a triangle?

The basic formula for the area of a triangle is: Area = 1/2 × base × height.

What are the three methods for calculating the area of a triangle using trigonometry?

The three methods for calculating the area of a triangle using trigonometry are: 1) When all three sides are known, using Heron's formula. 2) When two angles and one side are known, using a specific trigonometric formula involving sine. 3) When two sides and the included angle are known, using the formula 1/2 × a × b × sin(C).

What is Heron's formula used for in trigonometry?

Heron's formula is used to calculate the area of a triangle when the lengths of all three sides are known. The formula is: Area = √(s × (s - a) × (s - b) × (s - c)), where s is half of the perimeter of the triangle.

Why is the second method for calculating the area of a triangle not commonly used?

The second method, which involves the formula 1/2 × a² × sin(B) × sin(C) / sin(A), is rarely used because it requires a calculator to solve, making it more complex and less practical for manual calculations.

When should the third method (1/2 × a × b × sin(C)) be used?

The third method should be used when two sides of the triangle and the included angle are known. It is a simple and efficient formula for calculating the area of a triangle in this scenario.

In the first example, how do we calculate the semi-perimeter (s) of the triangle?

The semi-perimeter (s) is calculated by adding the three sides of the triangle and dividing the sum by 2. In the first example, the sides are 6 cm, 8 cm, and 4 cm, so s = (6 + 8 + 4) / 2 = 9 cm.

What does the term 's' represent in Heron's formula?

's' represents the semi-perimeter of the triangle, which is half of the perimeter. It is used in Heron's formula to calculate the area of a triangle when all three sides are known.

How do we calculate the area of a triangle when two sides and the angle between them are known?

When two sides (a and b) and the included angle (C) are known, the area of the triangle is calculated using the formula: Area = 1/2 × a × b × sin(C). This formula directly uses the sine of the included angle.

In the second example, how do we calculate the area of the triangle when two angles and one side are known?

In the second example, the side (c) is 12 cm, angle A is 60°, and angle B is 37°. To find the area, we first calculate angle C as 180° - 60° - 37° = 83°. Then, we apply the formula: Area = 1/2 × c² × sin(A) × sin(B) / sin(C), resulting in an area of 37.5 cm².

How do we solve for the length of CD in the fourth example, given the area of triangle ABC?

In the fourth example, the area of triangle ABC is given as 9, and we are asked to find the length of CD. We use the area formula for triangle ABD and relate it to triangle BCD. By replacing BD × sin(α) with CD and solving for CD, we find that CD = 6 cm.