- A right triangle is a special type of triangle in which the measure of one angle is exactly . The measure of the other two angles sums up to .
- Each side of a right triangle has a name, namely - hypotenuse, altitude (perpendicular), and base.
- The largest side of any right triangle is termed a hypotenuse and is always opposite to the right angle.
- The other two sides forming the right angle are called base and altitude (perpendicular).
- A right triangle is shown in the figure below.
- In the given right triangle , side opposite to the right angle is the hypotenuse.
- Notice that the altitude of a right triangle is defined relative to an angle. For example, relative to side AB acts as altitude. Relative to , side BC acts as altitude.
- The side left after assigning hypotenuse, and altitude is designated as a base.
- A right triangle obeys Pythagoras' theorem i.e .
- In a right-angled triangle, the ratios of sides concerning any of its acute angles are known as trigonometric ratios of that particular angle.
- In total, there are six trigonometric ratios.
- Consider right triangle ABC (as shown in the figure below) right angled at B, the trigonometric ratios of are defined as follows :
- The trigonometric ratios defined above are abbreviated as and respectively.
- A pair of angles whose sum is called complementary angles.
- In a right-angled triangle, the two acute angles are always complementary.
- Thus, for a right-angled triangle, as shown in the figure:
Example 1: Find the values of , and for the right triangle as shown in the figure.
Solution: For the angle '', , and . Thus,
Example 2: In a , right angled at B, if , then find the value of .
Solution: It is given that but . From here, we can conclude that and , where is any positive number. Let us now draw the triangle with the known data.
Using the Pythagorean theorem, we get,
Example 3: In a right triangle if , find the value of the acute angle .
Solution: It is given that . We can rewrite this expression as .
Example 4: Find the value of .
In any right-angled triangle, three basic trigonometric ratios defined for an acute angle '' are:
Relationship between trigonometric ratios:
- or or
- or or
- or or
- A triangle can never have two right angles.
- Trigonometric ratios apply to right-angled triangles only.