Welcome to trigonometry! Trigonometry is the study of relationships between angles and sides of triangles. In a right triangle, we have three sides: the hypotenuse, which is the longest side opposite to the right angle; the adjacent side, which is next to our angle of interest; and the opposite side, which is across from our angle. Trigonometry has numerous applications in navigation, engineering, physics, and computer graphics.
The three basic trigonometric functions are sine, cosine, and tangent. Sine is defined as the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. And tangent is the ratio of the opposite side to the adjacent side, which is also equal to sine divided by cosine. A helpful mnemonic to remember these ratios is SOH-CAH-TOA: Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent.
Another powerful way to understand trigonometric functions is through the unit circle. The unit circle is a circle with radius 1 centered at the origin of a coordinate system. When we place a point on this circle, its coordinates are directly related to the trigonometric functions. The x-coordinate of the point equals cosine theta, while the y-coordinate equals sine theta. Tangent theta is simply the ratio of y to x, or sine over cosine. As the angle theta increases, the point moves counterclockwise around the circle, and the sine and cosine values change accordingly.
The graphs of trigonometric functions reveal their periodic nature. The sine function, shown in red, starts at zero, increases to a maximum of 1 at pi over 2, returns to zero at pi, decreases to negative 1 at three pi over 2, and then returns to zero at two pi. The cosine function, shown in blue, follows a similar pattern but is shifted by pi over 2. It starts at 1, decreases to zero at pi over 2, reaches negative 1 at pi, increases back to zero at three pi over 2, and returns to 1 at two pi. Both functions repeat this pattern indefinitely, with a period of two pi. These periodic properties make trigonometric functions essential for modeling oscillatory phenomena in physics, engineering, and many other fields.
Trigonometric identities are equations that relate trigonometric functions to each other. The most fundamental identity is the Pythagorean identity: sine squared theta plus cosine squared theta equals one. Other important identities include the addition formulas, which express the sine or cosine of a sum or difference of angles, and the double angle formulas. These identities are powerful tools for simplifying expressions and solving problems. Trigonometry has countless applications across various fields. In physics, it's used to analyze waves and oscillations. Engineers use it for signal processing and electrical circuits. Navigation systems like GPS rely on trigonometric calculations to determine positions. And in computer graphics, trigonometry is essential for creating realistic 3D models and animations.