1.23: 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
"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?
Keep reading, and by the end of this Concept, you'll be able to help your friend.
Watch This
The middle portion of this video reviews the Quotient Identities.
James Sousa: The Reciprocal, Quotient, and Pythagorean Identities
Guidance
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
Example A
If
Solution:
Example B
Show that
Solution:
Example C
If
Solution:
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.
Guided Practice
1. If
2. If
3. If
Solutions:
1.
2.
3.
Concept Problem Solution
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:
Practice
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 oftanθ ?  If
sinθ=1237 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*}?
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Image Attributions
Here you'll learn what a quotient identity is and how to derive it.