1. Let's start by understanding what trigonometry is. Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. 2. The fundamental functions in trigonometry are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). They relate the angles of a right triangle to the ratios of its sides. 3. For a right triangle with an angle $\theta$, the sides are named as follows: the side opposite to $\theta$ is the opposite side, the side adjacent to $\theta$ is the adjacent side, and the longest side opposite the right angle is the hypotenuse. 4. The definitions of the trigonometric functions are: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 5. Important rules: - The values of $\sin$, $\cos$, and $\tan$ depend on the angle $\theta$. - These functions are periodic and have specific values at key angles like $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$. 6. Example: If a right triangle has an angle $\theta = 30^\circ$ and the hypotenuse is 10 units, find the length of the opposite side. 7. Using the sine function: $$\sin(30^\circ) = \frac{\text{opposite}}{10}$$ 8. We know $\sin(30^\circ) = 0.5$, so: $$0.5 = \frac{\text{opposite}}{10}$$ 9. Multiply both sides by 10: $$10 \times 0.5 = \cancel{10} \times \frac{\text{opposite}}{\cancel{10}}$$ $$5 = \text{opposite}$$ 10. Therefore, the opposite side is 5 units long. This is the basic introduction to trigonometry and how to use sine to find side lengths in right triangles.