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Quotient Identities

Tangent equals sine divided by cosine.

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Quotient Identities

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 \begin{align*}\frac{1}{2}\end{align*} for the sine, and \begin{align*}\frac{\sqrt{3}}{2}\end{align*} for the cosine. Why?" you ask.

"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:

\begin{align*}\sin \theta & = \frac{y}{r}\\ \cos \theta & = \frac{x}{r}\\ \tan \theta & = \frac{y}{x}\end{align*}

Given these definitions, we can show that \begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta}\end{align*}, as long as \begin{align*}\cos \theta \ne 0\end{align*}:

\begin{align*}\frac{\sin \theta}{\cos \theta} = \frac{\frac{y}{r}}{\frac{x}{r}} = \frac{y}{r} \times \frac{r}{x} = \frac{y}{x} = \tan \theta.\end{align*}

The equation \begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta}\end{align*} is therefore an identity that we can use to find the value of the tangent function, given the value of the sine and cosine.

Find the value of \begin{align*}\tan \theta\end{align*}?

If \begin{align*}\cos \theta = \frac{5}{13}\end{align*} and \begin{align*}\sin \theta = \frac{12}{13}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?

 \begin{align*}\tan \theta = \frac{12}{5}\end{align*}

\begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{13} \times \frac{13}{5} = \frac{12} {5}\end{align*}

Show that \begin{align*}\cot \theta = \frac{\cos \theta}{\sin \theta}\end{align*}

\begin{align*}\frac{\cos \theta}{\sin \theta} = \frac{\frac{x}{r}}{\frac{y}{r}} = \frac{x}{r} \times \frac{r}{y} = \frac{x}{y} = \cot \theta\end{align*}

What is the value of \begin{align*}\cot \theta\end{align*}?

If \begin{align*}\cos \theta = \frac{7}{25}\end{align*} and \begin{align*}\sin \theta = \frac{24}{25}\end{align*}, what is the value of \begin{align*}\cot \theta\end{align*}?

 \begin{align*}\cot \theta = \frac{7}{24}\end{align*}

\begin{align*}\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{7}{25}}{\frac{24}{25}} = \frac{7}{25} \times \frac{25}{24} = \frac{7} {24}\end{align*}

Examples

Example 1

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

Since you now know that:

\begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta}\end{align*}

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

\begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\end{align*}

Example 2

If \begin{align*}\cos \theta = \frac{17}{145}\end{align*} and \begin{align*}\sin \theta = \frac{144}{145}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?

 \begin{align*}\tan \theta = \frac{144}{17}\end{align*}. We can see this from the relationship for the tangent function:

\begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{144}{145}}{\frac{17}{145}} = \frac{144}{145} \times \frac{145}{17} = \frac{144} {17}\end{align*}

Example 3

 If \begin{align*}\sin \theta = \frac{63}{65}\end{align*} and \begin{align*}\cos \theta = \frac{16}{65}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?

\begin{align*}\tan \theta = \frac{63}{16}\end{align*}. We can see this from the relationship for the tangent function:

\begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{63}{65}}{\frac{16}{65}} = \frac{63}{65} \times \frac{65}{16} = \frac{63} {16}\end{align*}

Example 4

 If \begin{align*}\tan \theta = \frac{40}{9}\end{align*} and \begin{align*}\cos \theta = \frac{9}{41}\end{align*}, what is the value of \begin{align*}\sin \theta\end{align*}?

\begin{align*}\sin \theta = \frac{40}{41}\end{align*}. We can see this from the relationship for the tangent function:

\begin{align*}\tan \theta = \frac{\sin \theta}{\cos \theta}\\ \sin \theta = (\tan \theta)(\cos \theta)\\ \sin \theta = \frac{40}{9} \times \frac{9}{41}\\ \sin \theta = \frac{40}{41} \end{align*}

Review

Fill in each blank with a trigonometric function.

  1. \begin{align*}\tan \theta= \frac{\sin \theta}{?}\end{align*}
  2. \begin{align*}\cos \theta= \frac{\sin \theta}{?}\end{align*}
  3. \begin{align*}\cot \theta= \frac{?}{\sin \theta}\end{align*}
  4. \begin{align*}\cos \theta= (\cot \theta ) \cdot (?) \end{align*}
  5. If \begin{align*}\cos \theta = \frac{5}{13}\end{align*} and \begin{align*}\sin \theta = \frac{1}{13}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
  6. If \begin{align*}\sin \theta = \frac{3}{5}\end{align*} and \begin{align*}\cos \theta = \frac{4}{5}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
  7. If \begin{align*}\cos \theta = \frac{7}{25}\end{align*} and \begin{align*}\sin \theta = \frac{24}{25}\end{align*}, what is the value of \begin{align*}\tan \theta\end{align*}?
  8. 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*}?
  9. 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*}?
  10. 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*}?
  11. 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*}?
  12. 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*}?
  13. 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*}?
  14. 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*}?
  15. 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. 

Vocabulary

Quotient Identity

The quotient identity is an identity relating the tangent of an angle to the sine of the angle divided by the cosine of the angle.

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