Question

Define the six trigonometric functions in terms of the sides of right triangle.

Short answer

Question: Define the six trigonometric functions in terms of the sides of a right triangle. Answer: In the context of a right triangle, the six trigonometric functions are defined as follows: 1. sin(θ) = opposite/hypotenuse = b/c 2. cos(θ) = adjacent/hypotenuse = a/c 3. tan(θ) = opposite/adjacent = b/a 4. csc(θ) = hypotenuse/opposite = c/b 5. sec(θ) = hypotenuse/adjacent = c/a 6. cot(θ) = adjacent/opposite = a/b

Step-by-step solution

Label the sides of a right triangle

In a right triangle, the longest side (opposite the 90-degree angle) is called the hypotenuse. The other two sides can be referred to as the adjacent and opposite sides, depending on which angle we are considering. We will label the hypotenuse as \(c\), the adjacent side as \(a\), and the opposite side as \(b\). We will also use the variable \(\theta\) to denote the angle of interest.

Define sine, cosine, and tangent

The first three trigonometric functions are sin(\(\theta\)), cos(\(\theta\)), and tan(\(\theta\)). They are defined in terms of the sides of the right triangle as follows: 1. sine - sin(\(\theta\)): \(\frac{\text{opposite}}{\text{hypotenuse}} \Rightarrow \text{sin}(\theta)= \frac{b}{c}\) 2. cosine - cos(\(\theta\)): \(\frac{\text{adjacent}}{\text{hypotenuse}} \Rightarrow \text{cos}(\theta) = \frac{a}{c}\) 3. tangent - tan(\(\theta\)): \(\frac{\text{opposite}}{\text{adjacent}} \Rightarrow \text{tan}(\theta) = \frac{b}{a}\)

Define cosecant, secant, and cotangent

The next three trigonometric functions are reciprocal functions of the first three: cosecant (csc), secant (sec), and cotangent (cot). They are defined as follows: 4. cosecant - csc(\(\theta\)): \(\frac{1}{\text{sin}(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}} \Rightarrow \text{csc}(\theta) = \frac{c}{b}\) 5. secant - sec(\(\theta\)): \(\frac{1}{\text{cos}(\theta)}= \frac{\text{hypotenuse}}{\text{adjacent}} \Rightarrow \text{sec}(\theta) = \frac{c}{a}\) 6. cotangent - cot(\(\theta\)): \(\frac{1}{\text{tan}(\theta)}= \frac{\text{adjacent}}{\text{opposite}} \Rightarrow \text{cot}(\theta) = \frac{a}{b}\) With these definitions, we have represented all six trigonometric functions in terms of the sides of a right triangle.

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This right angle creates a perfect opportunity for understanding trigonometric functions. Here's how:

• **Hypotenuse**: This is the longest side of the right triangle. It is positioned opposite the right angle.
• **Opposite Side**: Depending on the angle of interest \(\theta\), the opposite side is the side that is directly opposite to \(\theta\).
• **Adjacent Side**: This is the side that forms the angle \(\theta\) with the hypotenuse, but is not the hypotenuse itself.

Understanding these sides helps establish relationships for each trigonometric function, which we will explore in the next sections.

Sine

The sine function, denoted as \( ext{sin}(\theta)\), is crucial in trigonometry. In a right triangle:

• It represents the ratio between the length of the side opposite the angle \(\theta\) and the hypotenuse.
• Mathematically, it is expressed as \( ext{sin}(\theta) = \frac{b}{c}\), where \(b\) is the length of the opposite side, and \(c\) is the hypotenuse.

This function allows you to understand how much the opposite side "wants" to point away or stretch away from the base, helping to calculate various scenarios in geometry and physics.

Cosine

Cosine, represented as \( ext{cos}(\theta)\), is another fundamental trigonometric function.

• This function is the ratio of the length of the adjacent side to the hypotenuse.
• The formula to remember is \( ext{cos}(\theta) = \frac{a}{c}\), where \(a\) is the length of the adjacent side.

Cosine explains the projection of any point along the base, giving insights into how much of the base is "covered" or "shadowed" by the angle's reach.

Tangent

Tangent, denoted as \( ext{tan}(\theta)\), is vital for understanding slopes and angles.

• It is the ratio of the opposite side's length to the adjacent side's length.
• Expressed mathematically, \( ext{tan}(\theta) = \frac{b}{a}\).

Tangent helps in calculating slopes and angles by revealing how steep a line is compared to a straight horizontal, similar to how a ramp or hill rises.