Trigonometric ratios are mathematical functions that relate the angles of a right triangle to the lengths of its sides. There are three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios are defined as follows:
- Sine (sin): The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse of the triangle.
- Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse of the triangle.
- Tangent (tan): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the adjacent side of the triangle.
These ratios are all interrelated, meaning that they can be expressed in terms of one another. For example, the Pythagorean theorem tells us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two other sides. Using this relationship, we can derive the following relationships between the trigonometric ratios:
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- cot²θ + 1 = csc²θ
Where θ is the measure of one of the acute angles in the triangle, sec is the secant ratio (the reciprocal of the cosine), and csc is the cosecant ratio (the reciprocal of the sine).
These relationships between the trigonometric ratios can be useful for solving problems in trigonometry, as well as for understanding the properties of right triangles and the relationships between their sides and angles.