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# Sine, Cosine, and Tangent Functions

## Trigonometric ratios based on sides of right triangles in relation to an angle.

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Sine, Cosine, and Tangent Functions

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?"

### Sine, Cosine and Tangent Functions

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 x\begin{align*}x\end{align*} in a right triangle, the sinx\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: sinA=ac\begin{align*}\sin A = \frac{a}{c}\end{align*} and sinB=bc\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: cosA=bc\begin{align*}\cos A = \frac{b}{c}\end{align*} and cosB=ac\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: tanA=ab\begin{align*}\tan A = \frac{a}{b}\end{align*} and tanB=ba\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 sinx,cosx\begin{align*}sin x, cos x\end{align*}, and tanx\begin{align*}tan x\end{align*} are just like f(x)\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 sinx\begin{align*}sin x\end{align*} it is still pronounced sine, with a long i. When we write cosx\begin{align*}cos x\end{align*}, we still say co-sine. And when we write tanx\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.

Use the definition of sine, cosine and tangent to find the values for angles in the given triangles.

Let's look at some example problems.

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

sinAcosAtanA=opposite sidehypotenuse=45=adjacent sidehypotenuse=35=opposite sideadjacent side=43\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*}

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

Using ABC:sinB=35\begin{align*}\triangle ABC: \sin B = \frac{3}{5}\end{align*}

Using NAP:sinB=610=35\begin{align*}\triangle NAP: \sin B = \frac{6}{10} = \frac{3}{5}\end{align*}

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

sinBtanA=opposite sidehypotenuse=1213=opposite sideadjacent side=512\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*}

### Examples

#### Example 1

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.

Using the triangle shown here:

#### Example 2

Find the sine of angle A\begin{align*}\angle{A}\end{align*}

The sine is equal to the opposite divided by the hypotenuse.

sinA=oppositehypotenuse=1161.18\begin{align*}\sin{A} = \frac{opposite}{hypotenuse} = \frac{11}{61} \approx .18\end{align*}

#### Example 3

Find the cosine of angle A\begin{align*}\angle{A}\end{align*}

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

cosA=adjacenthypotenuse=6061.98\begin{align*}\cos{A} = \frac{adjacent}{hypotenuse} = \frac{60}{61} \approx .98\end{align*}

#### Example 4

Find the tangent of angle A\begin{align*}\angle{A}\end{align*}

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

tanA=oppositeadjacent=61601.01\begin{align*}\tan{A} = \frac{opposite}{adjacent} = \frac{61}{60} \approx 1.01\end{align*}

### Review

Use the diagram below for questions 1-3.

1. Find sinA\begin{align*}\sin A\end{align*} and sinC\begin{align*}\sin C\end{align*}.
2. Find cosA\begin{align*}\cos A\end{align*} and cosC\begin{align*}\cos C\end{align*}.
3. Find tanA\begin{align*}\tan A\end{align*} and tanC\begin{align*}\tan C\end{align*}.

Use the diagram to fill in the blanks below.

1. tanA=??\begin{align*}\tan A = \frac{?}{?}\end{align*}
2. sinC=??\begin{align*}\sin C = \frac{?}{?}\end{align*}
3. tanC=??\begin{align*}\tan C = \frac{?}{?}\end{align*}
4. cosC=??\begin{align*}\cos C = \frac{?}{?}\end{align*}
5. sinA=??\begin{align*}\sin A = \frac{?}{?}\end{align*}
6. cosA=??\begin{align*}\cos A = \frac{?}{?}\end{align*}

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

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

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Color Highlighted Text Notes

### Vocabulary Language: English

cosine

The cosine of an angle in a right triangle is a value found by dividing the length of the side adjacent the given angle by the length of the hypotenuse.

sine

The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.