Advice 1: How to find the third side of a right triangle if the hypotenuse is known

The triangle is part of the plane bounded by three straight line segments, called the sides of the triangle in pairs having one common end called a vertex of the triangle. If one of the angles of a triangle is a straight line (90°), then the triangle is rectangular.
How to find the third side of a right triangle if the hypotenuse is known
Instruction
1
The sides of a right triangle adjacent to the right angle (AB and BC are called the legs. Side lying opposite the right angle is called the hypotenuse (AC).

Let we know the hypotenuse AC of the right triangle ABC: AC| = c. We denote the angle with vertex at point A as a ∟α, the angle with vertex at point B as ∟β. We need to find the lengths |AB| and |BC| of the other two sides.
2
Let known one of the legs of a right triangle. Assume |BC| = b. Then we can use the Pythagorean theorem that the hypotenuse squared is equal to the sum of the squares of the legs: a^2 + b^2 = c^2. From this equation we find unknown side |AB| = a = √ (c^2 - b^2).
3
Let you know one of the angles of a right triangle, suppose ∟α. Then the other two sides AB and BC of the triangle ABC can be found using trigonometric functions. So we get: sine ∟α equal to the ratio of the opposite leg to the hypotenuse sin α = b / c cosine ∟α equal to the ratio of adjacent sides to the hypotenuse cos α = a / c. Hence, we find the required lengths of the sides: |AB| = a = C * cos α, |BC| = b = c * sin α.
4
Let you know the ratio of sides k = a / b. Solve problems using trigonometric functions. The ratio a / b is none other than the cotangent ∟α: the ratio of the adjacent leg to the opposite ctg α = a / b. In this case, from this equality we Express a = b * ctg α. And substitute in the Pythagorean theorem a^2 + b^2 = c^2:

b^2 * ctg^2 α + b^2 = c^2. Make b^2 outside the brackets, we get b^2 * (ctg^2 α + 1) = c^2. And easily get the length of leg b = c / √(ctg^2 α + 1) = c / √(k^2 + 1), where k is a predetermined ratio of the legs.

By analogy, if you know the ratio of each b / a, we solve the problem with the use of trigonometric functions the tangent tg α = b / a. Substitute the value b = a * tg α in the Pythagorean theorem a^2 * tg^2 α + a^2 = c^2. Hence a = c / √(tg^2 α + 1) = c / √(k^2 + 1), where k is a predetermined ratio of the legs.
5
Consider special cases.

∟α = 30°. Then |AB| = a = c * cos α = c * √3 / 2; |BC| = b = c * sin α = c / 2.

∟α = 45°. Then |AB| = |BC| = a = b = c * √2 / 2.
Note
Square roots are extracted with a positive sign, because length cannot be negative. It seems obvious, but this error is very common, if to solve the problem automatically.
Useful advice
For finding the legs of a right triangle is convenient to use formula bring: sin β = sin (90° - α) = cos α; cos β = cos (90° - α) = sin α.

Advice 2: What are the sides of a right triangle

The amazing properties of right triangles people interested in antiquity. Many of these properties were described by Greek scientist Pythagoras. In Ancient Greece appeared, and the names of the sides of a right triangle.
What are the sides of a right triangle

What kind of triangle is called rectangular?


There are several types of triangles. Some have all the angles acute, the other one obtuse and two acute, the third – two sharp and straight. On this basis, each type of these geometric figures and received a name: acute-angled, obtuse-angled and rectangular. That is, rectangle is a triangle whose one angle is 90°. There is another definition, similar to the first. Is called a rectangular triangle whose two sides are perpendicular.

The hypotenuse and the legs


At acute and obtuse triangles to segments connecting the vertices of the angles, referred to as " the parties. From the rectangular triangle the parties have other names. Those that are adjacent to the right angle are called the legs. Side opposite the right angle is called the hypotenuse. In Greek the word "hypotenuse" means "stretched", and "side" - perpendicular.

The ratio between the hypotenuse and the legs of


The sides of a right triangle are linked by defined relationships, which greatly facilitates calculations. For example, knowing the sizes of the other two sides, you can calculate the length of the hypotenuse. This ratio is named opened his math called Pythagorean theorem, and it looks like this:

c2=a2+b2, where C is the hypotenuse, a and b are the legs. That is, the hypotenuse will equal the square root of the sum of the squares of the other two sides. To find any of the sides, enough of the square of the hypotenuse subtract the square of the other leg and removing from the obtained difference square root.

The adjacent and opposite side


Draw a right triangle ACB. Letter C to denote a vertex of a right angle, and the vertices of the acute angles. The side opposite each corner, conveniently called a, b and C, by the names in front of them reclining angles. Consider the angle A. the Leg and for him to be the opposite, side b – yard. The ratio of the opposite leg to the hypotenuse is called the sine. To calculate the trigonometric function by the formula: sinA=a/c. The ratio of adjacent sides to the hypotenuse is called the cosine. It is calculated by the formula: cosA=b/c.

Thus, knowing the angle and one side, you can use the following formulas to calculate the other side. Trigonometric ratios are connected both sides. The ratio of opposite over adjacent to is called the tangent, and adjacent to the opposite – cotangent. To Express these relationships can be a formula tgA=a/b or ctgA=b/a.
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