In geometry, a right triangle is a triangle with one right angle (90degree angle). The sides of a right triangle are commonly referred to as the hypotenuse, the adjacent side, and the opposite side. The hypotenuse is the longest side of the triangle and is opposite the right angle. The adjacent side is the side adjacent to the right angle and the opposite angle. The opposite side is the side opposite the right angle and the adjacent angle.
Calculating the sides of a right triangle is a common task in geometry and trigonometry. There are several methods for calculating the sides of a right triangle, depending on the information given. The most common methods involve using the Pythagorean theorem, trigonometric ratios, and special right triangles.
The Pythagorean theorem is a fundamental theorem in geometry that states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be used to calculate the length of the hypotenuse or the other sides of a right triangle.
Calculating Sides of Right Triangle
Calculating the sides of a right triangle involves using various methods and formulas.
 Pythagorean Theorem:
 Trigonometric Ratios:
 Sine, Cosine, Tangent:
 Special Right Triangles:
 306090 Triangle:
 454590 Triangle:
 Law of Sines:
 Law of Cosines:
These methods and formulas help determine the lengths of sides and angles in right triangles.
Pythagorean Theorem:
The Pythagorean theorem is a fundamental theorem in geometry that relates the squares of the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (the shorter sides).

Pythagorean Equation:
In a right triangle with sides a, b, and c, where c is the hypotenuse, the Pythagorean theorem can be expressed as:
$$c^2 = a^2 + b^2$$

Finding the Hypotenuse:
If you know the lengths of the other two sides (a and b), you can use the Pythagorean theorem to find the length of the hypotenuse (c) by squaring a and b, adding them together, and then taking the square root of the sum.

Finding the Other Sides:
If you know the length of the hypotenuse (c) and one of the other sides (a or b), you can use the Pythagorean theorem to find the length of the remaining side. Simply square the known side, subtract it from the square of the hypotenuse, and then take the square root of the difference.

Applications:
The Pythagorean theorem has numerous applications in various fields, including surveying, architecture, engineering, trigonometry, and navigation.
The Pythagorean theorem is a powerful tool for calculating the sides of right triangles, and it is essential for understanding many concepts in geometry and trigonometry.
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Sine, Cosine, Tangent:
In a right triangle, the trigonometric ratios sine, cosine, and tangent are defined as the ratios of the sides of the triangle. They are calculated using the following formulas:
 Sine (sin): $$\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$$
 Cosine (cos): $$\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$$
 Tangent (tan): $$\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}$$
Where θ is the angle opposite the side of interest.
These ratios can be used to find the lengths of the sides of a right triangle if you know the measure of one angle and one side. For example, if you know the measure of an angle and the length of the adjacent side, you can use the cosine ratio to find the length of the hypotenuse.
Trigonometric ratios are also used in a variety of applications, including navigation, surveying, and engineering.
In addition to the basic trigonometric ratios, there are also inverse trigonometric ratios, which are used to find the angle when given the ratio of two sides.
Special Right Triangles:
Certain right triangles, known as special right triangles, have specific angle and side relationships that make them useful for calculations. The most common special right triangles are the 306090 triangle and the 454590 triangle.
306090 Triangle:
 In a 306090 triangle, the angles are in a 30:60:90 ratio, with the right angle being 90 degrees.
 The side opposite the 30degree angle is half the length of the hypotenuse.
 The side opposite the 60degree angle is √3 times the length of the shorter side.
454590 Triangle:
 In a 454590 triangle, all three angles are 45 degrees.
 The two shorter sides are equal in length.
 The hypotenuse is √2 times the length of the shorter sides.
These special right triangles are particularly useful for finding side lengths and angles in other triangles, as well as for solving problems in trigonometry and geometry.
Other special right triangles include the 365390 triangle, the 453060 triangle, and the 603090 triangle. Each of these triangles has its own unique set of angle and side relationships.
306090 Triangle:
A 306090 triangle is a special right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. It has several remarkable properties and applications in geometry and trigonometry.
Properties of a 306090 Triangle:
 The side opposite the 30degree angle is half the length of the hypotenuse.
 The side opposite the 60degree angle is √3 times the length of the shorter side.
 The ratio of the shorter side to the longer side is 1:√3.
 The area of a 306090 triangle is given by the formula: $$Area = \frac{1}{2} \times \text{shorter side} \times \text{longer side}$$
Applications of a 306090 Triangle:
 Trigonometry: The 306090 triangle is used to define the trigonometric ratios of 30 degrees and 60 degrees.
 Navigation: The 306090 triangle is used in navigation to determine the direction and distance of a ship or aircraft.
 Surveying: The 306090 triangle is used in surveying to measure distances and angles.
 Architecture: The 306090 triangle is used in architecture to design buildings and structures with specific angles and proportions.
The 306090 triangle is a fundamental tool in geometry and trigonometry, and it has numerous applications in various fields.