Trigonometric functions are mathematical functions that describe the relationship between angles and sides in a right triangle. In a right triangle, we have angle theta, the opposite side b, the adjacent side a, and the hypotenuse c. These functions are fundamental tools used in mathematics, physics, and engineering.
The basic trigonometric functions include sine, cosine, and tangent. Sine theta is defined as the opposite side divided by the hypotenuse. Cosine theta is the adjacent side divided by the hypotenuse. Tangent theta is the opposite side divided by the adjacent side. These ratios help us calculate unknown sides and angles in triangles.
Trigonometric functions can also be understood through the unit circle. In the unit circle, angle theta corresponds to a point on the circle, where the x-coordinate is the cosine value and the y-coordinate is the sine value. As the point moves around the unit circle, the sine and cosine values change continuously. The tangent function equals the y-coordinate divided by the x-coordinate.
The graphs of trigonometric functions show their periodic nature. The sine function starts at zero, reaches one at pi over two, returns to zero at pi, goes to negative one at three pi over two, and back to zero at two pi. The cosine function follows a similar wave pattern but is shifted horizontally by pi over two. Both functions have a period of two pi, meaning they repeat their pattern every two pi units.
To summarize what we have learned: Trigonometric functions describe the fundamental relationships between angles and sides in triangles. The basic functions sine, cosine, and tangent can be understood through right triangles or the unit circle. These functions are periodic with period two pi and have wide applications in mathematics, physics, and engineering for solving problems involving angles, waves, and oscillations.