1. Let's start by stating the problem: We want to understand and derive the fundamental trigonometric equations related to triangles. 2. Consider a right triangle with an angle $\theta$. The sides are labeled as follows: the side opposite $\theta$ is $a$, the adjacent side is $b$, and the hypotenuse is $c$. 3. The primary trigonometric functions are defined as ratios of these sides: - Sine: $\sin(\theta) = \frac{a}{c}$ - Cosine: $\cos(\theta) = \frac{b}{c}$ - Tangent: $\tan(\theta) = \frac{a}{b}$ 4. From these definitions, we can derive the Pythagorean identity. Since $c$ is the hypotenuse, by the Pythagorean theorem: $$a^2 + b^2 = c^2$$ Dividing both sides by $c^2$ gives: $$\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = 1$$ Which means: $$\sin^2(\theta) + \cos^2(\theta) = 1$$ 5. Next, we derive the tangent identity using sine and cosine: $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$ 6. For the other trigonometric functions: - Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{c}{a}$ - Secant: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{c}{b}$ - Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{b}{a}$ 7. These identities are fundamental in solving problems involving triangles and oscillatory phenomena. Final answer: The key triangle trigonometric equations are: $$\sin(\theta) = \frac{a}{c}, \quad \cos(\theta) = \frac{b}{c}, \quad \tan(\theta) = \frac{a}{b}$$ and the Pythagorean identity: $$\sin^2(\theta) + \cos^2(\theta) = 1$$