Understanding the Pythagorean Theorem and Trigonometry

The Pythagorean Theorem is a big idea in math that keeps showing up in different areas. It also has a strong connection to trigonometry, which is the study of triangles.

At its simplest, the theorem tells us how the sides of a right triangle relate to each other. It says that if we take the longest side, called the hypotenuse (let's call it $c$), and square its length, we get the same result as adding the squares of the other two sides ($a$ and $b$). We can write this as:

$$c^2 = a^2 + b^2$$

Now, let's add trigonometry into the mix. In a right triangle, we can use two important functions called sine and cosine.

Here’s how they work:

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c}$
• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c}$

These definitions help us understand how the sides relate to the angles.

If we take these formulas and square both sides, we get something interesting:

$$\sin^2(\theta) + \cos^2(\theta) = \frac{b^2}{c^2} + \frac{a^2}{c^2} = \frac{a^2 + b^2}{c^2} = 1$$

This shows us that the Pythagorean Theorem connects with trigonometry. No matter what angle you have in a right triangle, the relationship among the sides stays the same. In other words, $a^2 + b^2 = c^2$, and this idea works with trigonometric functions too.

When we draw these functions on graphs, especially using the unit circle, things become clearer. On the circle, every point has coordinates $(\cos(\theta), \sin(\theta))$. Since the radius (the hypotenuse $c$) is always 1 on this circle, the equation stays true.

This blend of geometry and trigonometry helps us understand triangles better and makes math feel more connected.