In this lesson, we will introduce you to the Pythagorean theorem and its applications in trigonometry.

Pythagorean Theorem: The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides (the adjacent and opposite sides).

In mathematical terms, it can be expressed as:

c^2 = a^2 + b^2

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Applications: The Pythagorean theorem has many applications in various fields such as architecture, engineering, and physics. Some of the common applications are:

- Finding the length of a missing side of a right-angled triangle: The Pythagorean theorem can be used to find the length of a missing side of a right-angled triangle when the lengths of the other two sides are known.
- Checking if a triangle is a right-angled triangle: If the lengths of the three sides of a triangle satisfy the Pythagorean theorem, then it is a right-angled triangle.
- Finding the shortest distance between two points: The Pythagorean theorem can be used to find the shortest distance between two points in a two-dimensional space.
- Calculating the distance between two points in a three-dimensional space: The Pythagorean theorem can be extended to calculate the distance between two points in a three-dimensional space.

Example: Suppose we have a right-angled triangle with the adjacent side (a) measuring 3 units and the opposite side (b) measuring 4 units. We can use the Pythagorean theorem to find the length of the hypotenuse (c).

c^2 = a^2 + b^2 c^2 = 3^2 + 4^2 c^2 = 9 + 16 c^2 = 25 c = √25 c = 5

Therefore, the length of the hypotenuse is 5 units.

Conclusion: In conclusion, the Pythagorean theorem is a fundamental concept in trigonometry that is widely used in various fields such as architecture, engineering, and physics. It states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean theorem has many applications, including finding the length of a missing side of a right-angled triangle, checking if a triangle is a right-angled triangle, finding the shortest distance between two points, and calculating the distance between two points in a three-dimensional space.