You are working in math class one day when your friend leans over and asks you what you got for the sine and cosine of a particular angle.

"I got

"It looks like I'm supposed to calculate the tangent function for the same angle you just did, but I can't remember the relationship for tangent. What should I do?" he says.

Do you know how you can help your friend find the answer, even if both you and he don't remember the relationship for tangent?

### Quotient Identities

The definitions of the trig functions led us to the reciprocal identities, which can be seen in the Concept about that topic. They also lead us to another set of identities, the quotient identities.

Consider first the sine, cosine, and tangent functions. For angles of rotation (not necessarily in the unit circle) these functions are defined as follows:

Given these definitions, we can show that

The equation

Let's take a look at some problems involving quotient identities.

1. Find the value of

If

2. Show that

3. What is the value of

If

### Examples

#### Example 1

Earlier, you were asked if you can help your friend find the answer.

Since you now know that:

you can use this knowledge to help your friend with the sine and cosine values you measured for yourself earlier:

#### Example 2

If

#### Example 3

If

#### Example 4

If

### Review

Fill in each blank with a trigonometric function.

tanθ=sinθ? cosθ=sinθ? cotθ=?sinθ cosθ=(cotθ)⋅(?) - If
cosθ=513 andsinθ=113 , what is the value oftanθ ? - If
sinθ=35 andcosθ=45 , what is the value oftanθ ? - If
cosθ=725 andsinθ=2425 , what is the value of \begin{align*}\tan \theta\end{align*}? - If \begin{align*}\sin \theta = \frac{12}{37}\end{align*} and \begin{align*}\cos \theta = \frac{35}{37}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
- If \begin{align*}\cos \theta = \frac{20}{29}\end{align*} and \begin{align*}\sin \theta = \frac{21}{29}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
- If \begin{align*}\sin \theta = \frac{39}{89}\end{align*} and \begin{align*}\cos \theta = \frac{80}{89}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
- If \begin{align*}\cos \theta = \frac{48}{73}\end{align*} and \begin{align*}\sin \theta = \frac{55}{73}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
- If \begin{align*}\sin \theta = \frac{65}{97}\end{align*} and \begin{align*}\cos \theta = \frac{72}{97}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
- If \begin{align*}\cos \theta = \frac{1}{2}\end{align*} and \begin{align*}\cot \theta = \frac{\sqrt{3}}{3}\end{align*}, what is the value of \begin{align*}\sin \theta\end{align*}?
- If \begin{align*}\tan \theta = 0\end{align*} and \begin{align*}\cos \theta = -1\end{align*}, what is the value of \begin{align*}\sin \theta\end{align*}?
- If \begin{align*}\cot \theta = -1\end{align*} and \begin{align*}\sin \theta = -\frac{\sqrt{2}}{2}\end{align*}, what is the value of \begin{align*}\cos \theta\end{align*}?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.23.