Welcome to trigonometry! Trigonometry is the study of relationships between angles and sides in triangles. It's particularly useful when working with right triangles. In a right triangle, we have three main parts: the hypotenuse, which is the longest side opposite the right angle, the adjacent side next to our angle theta, and the opposite side across from our angle.
The three main trigonometric ratios are sine, cosine, and tangent. Sine equals opposite over hypotenuse. Cosine equals adjacent over hypotenuse. Tangent equals opposite over adjacent. We remember this with the mnemonic SOH-CAH-TOA: Sine Opposite Hypotenuse, Cosine Adjacent Hypotenuse, Tangent Opposite Adjacent.
There are three special angles that appear frequently in trigonometry: thirty, forty-five, and sixty degrees. For thirty degrees, sine equals one half, cosine equals square root of three over two, and tangent equals one over square root of three. For forty-five degrees, sine and cosine both equal one over square root of two, and tangent equals one. For sixty degrees, sine equals square root of three over two, cosine equals one half, and tangent equals square root of three. These values are essential to memorize!
Let's solve a practical problem. We have a right triangle where the angle is thirty degrees and the hypotenuse is ten units. We need to find the opposite side. Using sine, we write sine of thirty degrees equals opposite over ten. Since sine of thirty degrees is one half, we get one half equals opposite over ten. Multiplying both sides by ten, we find the opposite side equals five units.
Trigonometry has countless real-world applications. We use it to find heights of buildings using angle of elevation. GPS and navigation systems rely on trigonometric calculations. Engineers use it in construction and design. Physicists apply it to analyze waves and oscillations. Even computer graphics and video games use trigonometry for realistic animations. Remember the formula: tangent of the angle equals height over distance. Master these trigonometric basics, and you'll be equipped to solve many practical problems in science, technology, and everyday life!