Trigonometry is a fundamental branch of mathematics that explores the relationships between the sides and angles of triangles. It focuses particularly on right triangles, where one angle is exactly 90 degrees. In this triangle, we can see angle theta and the three sides labeled a, b, and c.
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. Tangent is the ratio of the opposite side to the adjacent side. These ratios help us calculate unknown sides and angles in right triangles.
Trigonometric functions can also be understood using the unit circle. In a unit circle with radius one, any angle theta corresponds to a point on the circle. The x-coordinate of this point equals cosine theta, and the y-coordinate equals sine theta. As the point moves around the circle, these values change continuously, showing how trigonometric functions behave for all angles.
The graphs of trigonometric functions reveal their periodic nature. The sine function creates a smooth wave that starts at zero, rises to one at pi over two, returns to zero at pi, drops to negative one at three pi over two, and completes the cycle at two pi. The cosine function follows a similar pattern but starts at one. Both functions repeat this pattern indefinitely, making them essential for describing oscillations and waves in physics and engineering.
Trigonometry has countless real-world applications. It's essential in navigation and GPS systems, engineering and construction projects, physics for analyzing waves and oscillations, computer graphics for rendering, music for understanding sound waves, and astronomy for calculating distances to stars. For example, surveyors use trigonometry to measure building heights by observing the angle of elevation and knowing the horizontal distance. The tangent function relates the angle to the ratio of height over distance, making such calculations possible.