Define the sine function in terms of a right triangle.

The **sine function** is one of the fundamental trigonometric functions and can be easily understood using a **right triangle**. Here's how the sine of an angle is defined in the context of a right triangle: ### Right Triangle Basics A **right triangle** is a triangle that has one angle exactly equal to 90 degrees. The sides of a right triangle are typically labeled as follows: - **Hypotenuse** (`c`): The side opposite the right angle; it is the longest side of the triangle. - **Opposite side** (`a`): The side opposite the angle you are considering (let's call this angle θ). - **Adjacent side** (`b`): The side adjacent (next to) the angle θ, excluding the hypotenuse. ### Defining Sine in a Right Triangle For a given acute angle θ in a right triangle, the sine of θ (written as sin θ) is defined as the ratio of the length of the **opposite side** to the length of the **hypotenuse**. Mathematically, this is expressed as: \[ \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{a}{c} \] ### Visual Representation Consider the following right triangle: ``` |\ | \ a | \ c | \ |____\ b ``` - Angle θ is one of the non-right angles. - Side `a` is opposite angle θ. - Side `b` is adjacent to angle θ. - Side `c` is the hypotenuse. In this triangle: \[ \sin \theta = \frac{a}{c} \] ### Example Suppose you have a right triangle where: - The length of the hypotenuse (`c`) is 10 units. - The length of the side opposite angle θ (`a`) is 6 units. To find sin θ: \[ \sin \theta = \frac{a}{c} = \frac{6}{10} = 0.6 \] So, **sin θ = 0.6**. ### Summary - **Sine of an angle (sin θ)** in a right triangle is the ratio of: - **Opposite side** (the side opposite the angle) - **Divided by the hypotenuse** (the longest side of the triangle) Understanding this ratio is essential for solving various problems in trigonometry, such as finding unknown side lengths or angles in right triangles.