Sine and cosine are fundamental trigonometric functions that help us understand the relationship between angles and sides in triangles. In a right triangle, we have an angle theta, the opposite side, the adjacent side, and the hypotenuse. These functions form the foundation of trigonometry.
The mnemonic SOH CAH TOA helps us remember the definitions. SOH means Sine equals Opposite over Hypotenuse. CAH means Cosine equals Adjacent over Hypotenuse. In our triangle, sine of theta is the opposite side divided by the hypotenuse, while cosine of theta is the adjacent side divided by the hypotenuse.
The unit circle extends our understanding beyond right triangles. On a circle with radius one, any angle theta corresponds to a point with coordinates cosine theta and sine theta. The cosine is the x-coordinate and the sine is the y-coordinate. This definition works for any angle, not just acute angles.
The graphs of sine and cosine reveal their periodic nature. Both functions repeat every 2 pi radians and have values between negative 1 and positive 1. The sine function starts at zero and increases, while the cosine function starts at 1 and decreases. Notice how cosine is just sine shifted by pi over 2.
In summary, sine and cosine are fundamental functions that relate angles to ratios. We learned the triangle definitions using SOH CAH TOA, extended them to the unit circle where they represent coordinates, and saw their periodic nature in graphs. The fundamental identity sine squared plus cosine squared equals one connects these concepts beautifully.