Trigonometric Ratios in Right-Angled Triangles

Trigonometry is a branch of mathematics that deals with the relationship between the angles and sides of a right-angled triangle. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). In addition, there are three more ratios called cotangent (cot), secant (sec), and cosecant (cosec). The trigonometric ratios are defined as the values of all trigonometric functions based on the ratio of sides in a right-angled triangle. For example, sine is defined as the ratio of the side opposite to an angle to the hypotenuse, cosine is the ratio of the side adjacent to an angle to the hypotenuse, and tangent is the ratio of the side opposite to the side adjacent.

To find the trigonometric ratios, you can use the following formulas:

• Sine (sin): sin(θ) = opposite side / hypotenuse
• Cosine (cos): cos(θ) = adjacent side / hypotenuse
• Tangent (tan): tan(θ) = opposite side / adjacent side
• Cotangent (cot): cot(θ) = adjacent side / opposite side
• Secant (sec): sec(θ) = hypotenuse / adjacent side
• Cosecant (cosec): cosec(θ) = hypotenuse / opposite side

These ratios can be used to solve problems involving angles and sides in right-angled triangles. For example, given a right triangle with a hypotenuse of 5 cm, a base of 3 cm, and a perpendicular of 4 cm, you can find the trigonometric ratios tan(θ), sin(θ), and cos(θ) using the formulas above.

Trigonometric ratios have a wide range of applications in everyday life, such as in navigation, engineering, astronomy, and physics. They are also used in solving problems involving angles and sides in right-angled triangles, which are a fundamental concept in geometry.