Trigonometry is a branch of mathematics that studies triangles, especially right-angled triangles. These are triangles that have one angle measuring exactly ninety degrees. Trigonometry helps us understand the relationship between the angles and the side lengths in these special triangles.
The three main trigonometric functions are sine, cosine, and tangent. These functions help us find missing sides and angles in right triangles. Each function relates a specific angle to the ratios of different sides of the triangle. The opposite side is across from the angle, the adjacent side is next to the angle, and the hypotenuse is the longest side opposite the right angle.
The trigonometric functions are defined as ratios of the sides of a right triangle. Sine theta equals opposite over hypotenuse. Cosine theta equals adjacent over hypotenuse. Tangent theta equals opposite over adjacent. These ratios stay the same for any angle, regardless of the triangle size. For example, if we have a triangle with opposite side 2, adjacent side 3, and hypotenuse 3.6, we can calculate sine as 0.56, cosine as 0.83, and tangent as 0.67.
Trigonometry helps us measure things indirectly in many real-world situations. We can find the height of buildings, measure distances across rivers, use it in navigation and GPS systems, engineering and construction, and even computer graphics and games. For example, to find a tree's height, we can measure our distance from the tree and the angle to the top, then use trigonometry. If we stand 20 meters away and look up at a 30-degree angle, we can use tangent to calculate the height.
To summarize what we've learned: Trigonometry is a branch of mathematics that studies right triangles and the relationships between their angles and sides. The three main functions - sine, cosine, and tangent - are ratios that help us find missing parts of triangles. This powerful tool enables us to measure things indirectly in the real world and is used in navigation, engineering, and many other important fields.