Welcome to trigonometry! Trigonometry is the mathematical study of triangles and the relationships between their angles and sides. This fundamental branch of mathematics has countless real-world applications, from navigation systems and engineering to computer graphics and astronomy. Let's explore this fascinating subject together.
Now let's explore the three fundamental trigonometric functions. 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. These ratios are constant for any given angle, making them incredibly useful for solving problems involving triangles.
The unit circle is a powerful tool that extends trigonometry beyond right triangles. It's a circle with radius 1 centered at the origin. Any point on this circle has coordinates cosine theta, sine theta. The x-coordinate gives us the cosine value, and the y-coordinate gives us the sine value. Watch as the point moves around the circle and see how these values change.
Trigonometric functions create beautiful wave patterns when graphed. The sine function starts at the origin and oscillates between negative 1 and positive 1 with a period of 2 pi. The cosine function has the same shape but starts at 1 instead of 0. Notice how cosine is essentially sine shifted to the left by pi over 2 units. These periodic patterns appear everywhere in nature, from sound waves to ocean tides.
To summarize what we've learned about trigonometry: It's the study of relationships between angles and sides in triangles. The three basic functions sine, cosine, and tangent provide these relationships. The unit circle extends these concepts beyond right triangles. Trigonometric graphs reveal beautiful periodic patterns. These fundamental concepts have countless applications across science, engineering, and technology.