Can you explain me trigonometric formulas and how they work?

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Welcome to trigonometry! Trigonometric formulas describe the relationships between angles and sides of triangles. In a right triangle, we define three basic ratios relative to an angle theta. Sine equals the opposite side divided by the hypotenuse. Cosine equals the adjacent side divided by the hypotenuse. And tangent equals the opposite side divided by the adjacent side. These ratios are the foundation of all trigonometric formulas. The unit circle is a powerful way to extend trigonometric functions beyond right triangles to all angles. It's a circle with radius 1 centered at the origin of a coordinate system. For any angle theta, measured counterclockwise from the positive x-axis, the point where the angle's terminal side intersects the unit circle has coordinates cosine theta and sine theta. This means the x-coordinate equals cosine theta, and the y-coordinate equals sine theta. Tangent theta equals sine theta divided by cosine theta, which is the y-coordinate divided by the x-coordinate. As the angle changes, these values change periodically, creating the familiar trigonometric wave patterns. Trigonometric functions can be visualized as graphs, which reveal their periodic nature. The sine function, shown in blue, starts at zero, increases to one at pi over two, returns to zero at pi, decreases to negative one at three pi over two, and returns to zero at two pi. The cosine function, shown in red, follows a similar pattern but is shifted by pi over two. It starts at one, decreases to zero at pi over two, reaches negative one at pi, increases back to zero at three pi over two, and returns to one at two pi. Both functions repeat this pattern indefinitely, with a period of two pi. These wave-like patterns are fundamental in modeling oscillatory phenomena in physics, engineering, and many other fields. Trigonometric identities are equations that relate different trigonometric functions and are true for all values where the functions are defined. The most fundamental identity is the Pythagorean identity: sine squared theta plus cosine squared theta equals one. This can be visualized using the unit circle, where the point at angle theta has coordinates cosine theta and sine theta. By the Pythagorean theorem, the square of these coordinates must sum to one, which is the square of the radius. Another basic identity is the quotient identity: tangent theta equals sine theta divided by cosine theta. The reciprocal identities define cosecant, secant, and cotangent as the reciprocals of sine, cosine, and tangent respectively. These identities are essential for simplifying expressions, solving equations, and deriving more complex formulas. To summarize what we've learned about trigonometric formulas: Trigonometric functions fundamentally describe the relationships between angles and sides in triangles. The basic ratios—sine, cosine, and tangent—are defined using right triangles as ratios of sides. The unit circle extends these definitions to all angles, not just those in right triangles. All trigonometric functions are periodic, repeating their values every 2π radians or 360 degrees. Finally, trigonometric identities provide powerful relationships between different functions, allowing us to simplify expressions and solve complex problems. These concepts form the foundation of trigonometry and have applications in numerous fields including physics, engineering, computer graphics, and signal processing.