You are helping your grandfather with some repairs around the house when he mentions that he could use some help painting some boards on the staircase of his front porch. When you go out to see what he means, you notice that the stairs are supported by a set of boards that are glued together so that they are shaped like a triangle, with one extra board placed over top of them for decoration.

As you are looking around for paint, you think about the situation and realize that this reminds you a lot of your math class. In your study of triangles, you are about to start a unit working with the relationships between the sides of a triangle. You begin to wonder: "How many possible relationships are there between sides of triangles, anyway?"

By the end of this Concept, you'll have studied three of these important relationships, as well as know how many relationships there are total.

### Watch This

James Sousa: The Trigonometric Functions in Terms of Right Triangles

### Guidance

The first three trigonometric functions we will work with are the sine, cosine, and tangent functions. The elements of the domains of these functions are angles. We can define these functions in terms of a right triangle: The elements of the range of the functions are particular ratios of sides of triangles.

We define the sine function as follows: For an acute angle \begin{align*}x\end{align*} in a right triangle, the \begin{align*}sin x\end{align*} is equal to the ratio of the side opposite of the angle over the hypotenuse of the triangle. For example, using this triangle, we have: \begin{align*}\sin A = \frac{a}{c}\end{align*} and \begin{align*}\sin B = \frac{b}{c}\end{align*}.

Since all right triangles with the same acute angles are similar, this function will produce the same ratio, no matter which triangle is used. Thus, it is a well-defined function.

Similarly, the cosine of an angle is defined as the ratio of the side adjacent (next to) the angle over the hypotenuse of the triangle. Using this triangle, we have: \begin{align*}\cos A = \frac{b}{c}\end{align*} and \begin{align*}\cos B = \frac{a}{c}\end{align*}.

Finally, the tangent of an angle is defined as the ratio of the side opposite the angle to the side adjacent to the angle. In the triangle above, we have: \begin{align*}\tan A = \frac{a}{b}\end{align*} and \begin{align*}\tan B = \frac{b}{a}\end{align*}.

There are a few important things to note about the way we write these functions. First, keep in mind that the abbreviations \begin{align*}sin x, cos x\end{align*}, and \begin{align*}tan x\end{align*} are just like \begin{align*}f(x)\end{align*}. They simply stand for specific kinds of functions. Second, be careful when using the abbreviations that you still pronounce the full name of each function. When we write \begin{align*}sin x\end{align*} it is still pronounced *sine*, with a long *i.* When we write \begin{align*}cos x\end{align*}, we still say co-sine. And when we write \begin{align*}tan x\end{align*}, we still say tangent.

We can use these definitions to find the sine, cosine, and tangent values for angles in a right triangle.

#### Example A

Find the sine, cosine, and tangent of \begin{align*}\angle{A}\end{align*}:

**Solution:**

\begin{align*}\sin A & = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{4}{5}\\ \cos A & = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{3}{5}\\ \tan A & = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{4}{3}\end{align*}

#### Example B

Find \begin{align*}\sin B\end{align*} using \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle NAP.\end{align*}

**Solution:**

Using \begin{align*}\triangle ABC: \sin B = \frac{3}{5}\end{align*}

Using \begin{align*}\triangle NAP: \sin B = \frac{6}{10} = \frac{3}{5}\end{align*}

#### Example C

Find \begin{align*}\sin B\end{align*} and \begin{align*}\tan A\end{align*} in the triangle below:

**Solution:**

\begin{align*}\sin B & = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{12}{13}\\ \tan A & = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{5}{12}\end{align*}

### Guided Practice

Using the triangle shown here:

Find

1. The sine of angle \begin{align*}\angle{A}\end{align*}

2. The cosine of angle \begin{align*}\angle{A}\end{align*}

3. The tangent of angle \begin{align*}\angle{A}\end{align*}

**Solution:** 1. The sine is equal to the opposite divided by the hypotenuse.

\begin{align*}\sin{A} = \frac{opposite}{hypotenuse} = \frac{11}{61} \approx .18\end{align*}

2. The cosine is equal to the adjacent divided by the hypotenuse.

\begin{align*}\cos{A} = \frac{adjacent}{hypotenuse} = \frac{60}{61} \approx .98\end{align*}

3. The tangent is equal to the opposite divided by the adjacent.

\begin{align*}\tan{A} = \frac{opposite}{adjacent} = \frac{61}{60} \approx 1.01\end{align*}

### Concept Problem Solution

Looking at a triangle like the shape of your grandfather's staircase:

We can see that there are several ways to make relationships between the sides. In this case, we are only interested in ratios between the sides, which means one side will be divided by another. If we assume that dividing a side by itself doesn't count (since the answer would always be equal to one), let's look at the number of possible combinations:

If we use the angle labelled above, there is: 1) The side opposite the angle divided by the hypotenuse (the sine function) 2) The side adjacent the angle divided by the hypotenuse (the cosine function) 3) The side opposite the angle divided by adjacent side (the tangent function)

You can also imagine taking the same sides, except reversing the numerator and denominator: 4) The hypotenuse divided by the side opposite the angle 5) The hypotenuse divided by the side adjacent to the angle 6) The side adjacent to the angle divided by the side opposite to the angle

The first three functions are what we introduced in this Concept. The last three are other functions you'll learn about in a different Concept.

### Explore More

Use the diagram below for questions 1-3.

- Find \begin{align*}\sin A\end{align*} and \begin{align*}\sin C\end{align*}.
- Find \begin{align*}\cos A\end{align*} and \begin{align*}\cos C\end{align*}.
- Find \begin{align*}\tan A\end{align*} and \begin{align*}\tan C\end{align*}.

Use the diagram to fill in the blanks below.

- \begin{align*}\tan A = \frac{?}{?}\end{align*}
- \begin{align*}\sin C = \frac{?}{?}\end{align*}
- \begin{align*}\tan C = \frac{?}{?}\end{align*}
- \begin{align*}\cos C = \frac{?}{?}\end{align*}
- \begin{align*}\sin A = \frac{?}{?}\end{align*}
- \begin{align*}\cos A = \frac{?}{?}\end{align*}

From questions 4-9, we can conclude the following. Fill in the blanks.

- \begin{align*}\cos \underline{\;\;\;\;\;\;\;} = \sin A\end{align*} and \begin{align*}\sin \underline{\;\;\;\;\;\;\;} = \cos A\end{align*}.
- \begin{align*}\tan A\end{align*} and \begin{align*}\tan C\end{align*} are _________ of each other.
- Explain why the cosine of an angle will never be greater than 1.
- Use your knowledge of 45-45-90 triangles to find the sine, cosine, and tangent of a 45 degree angle.
- Use your knowledge of 30-60-90 triangles to find the sine, cosine, and tangent of a 30 degree angle.
- Use your knowledge of 30-60-90 triangles to find the sine, cosine, and tangent of a 60 degree angle.
- As the degree of an angle increases, will the tangent of the angle increase or decrease? Explain.