Geometry/Right Triangles and Pythagorean Theorem

Right triangles
Right triangles are triangles in which one of the interior angles is 90o. A 90o angle is called a right angle. Right triangles have special properties which make it easier to conceptualize and calculate their parameters in many cases.

The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are the legs. When using the Pythagorean Theorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.



Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula:

$$Area = \,$$(1/2)$$ b h \, $$

This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b &times; h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle.



Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45&deg; angles as well as the 90&deg; angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles (see ../Triangle/), then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagorean theorem can be used for non right triangles. a2+b2=c2-2c

Pythagorean Theorem
For history regarding the Pythagorean Theorem, see Pythagorean theorem. The Pythagorean Theorem states that:


 * In a right 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.

Let's take a right triangle as shown here and set c equal to the length of the hypotenuse and set a and b each equal to the lengths of the other two sides. Then the Pythagorean Theorem can be stated as this equation:

$$ \quad c^2 = a^2 + b^2 $$

Using the Pythagorean Theorem, if the lengths of any two of the sides of a right triangle are known and it is known which side is the hypotenuse, then the length of the third side can be determined from the formula.

Sine, Cosine, and Tangent for Right Triangles
Sine, Cosine, and Tangent are all functions of an angle, which are useful in right triangle calculations. For an angle designated as &theta;, the sine function is abbreviated as sin &theta;, the cosine function is abbreviated as cos &theta;, and the tangent function is abbreviated as tan &theta;. For any angle &theta;, sin &theta;, cos &theta;, and tan &theta; are each single determined values and if &theta; is a known value, sin &theta;, cos &theta;, and tan &theta; can be looked up in a table or found with a calculator. There is a table listing these function values at the end of this section. For an angle between listed values, the sine, cosine, or tangent of that angle can be estimated from the values in the table. Conversely, if a number is known to be the sine, cosine, or tangent of an angle, then such tables could be used in reverse to find (or estimate) the value of a corresponding angle.

These three functions are related to right triangles in the following ways:

In a right triangle,


 * the sine of a non-right angle equals the length of the leg opposite that angle divided by the length of the hypotenuse.


 * the cosine of a non-right angle equals the length of the leg adjacent to it divided by the length of the hypotenuse.


 * the tangent of a non-right angle equals the length of the leg opposite that angle divided by the length of the leg adjacent to it.

For any value of &theta; where cos &theta; &ne; 0,

$$\qquad \tan \theta = \frac {\sin \theta } {\cos \theta }$$.

If one considers the diagram representing a right triangle with the two non-right angles &theta;1and &theta;2, and the side lengths a,b,c as shown here:



For the functions of angle &theta;1:

$$\sin \theta_1 = \frac {b} {c} \qquad \cos \theta_1 = \frac{a} {c} \qquad \tan \theta_1 = \frac {b} {a} $$

Analogously, for the functions of angle &theta;2:

$$\sin \theta_2 = \frac {a} {c} \qquad \cos \theta_2 = \frac {b} {c} \qquad \tan \theta_2 = \frac {a} {b} $$

Table of sine, cosine, and tangent for angles &theta; from 0 to 90&deg;
General rules for important angles: $$ sin 45^o = cos 45^o = \frac{1}{\sqrt{2}} $$

$$ tan 45^o = 1 $$

$$ sin 30^o = \frac{1}{2} = cos 60^o $$

$$ cos 30^o = \frac{\sqrt{3}}{2} = sin 60^o $$