Wize High School Algebra II Textbook (Common Core) > Trigonometric Identities & Proofs

Reciprocal & Quotient Identities

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Reciprocal & Quotient Identities
The following trigonometric ratios are known as reciprocal identities:

sin⁡θ=1csc⁡θcos⁡θ=1sec⁡θtan⁡θ=1cot⁡θ\begin{array}{ccc}
\sin{\theta}=\displaystyle\frac{1}{\csc\theta}&&&\cos{\theta}=\displaystyle\frac{1}{\sec\theta}&&&\tan{\theta}=\displaystyle\frac{1}{\cot{\theta}}
\end{array}sinθ=cscθ1​​​​cosθ=secθ1​​​​tanθ=cotθ1​​

Example

If sin⁡θ=12\sin\theta=\displaystyle\frac{1}{2}sinθ=21​, then csc⁡θ=2\csc{\theta}=2cscθ=2.

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The following trigonometric ratios are known as quotient identities:

tan⁡θ=sin⁡θcos⁡θcot⁡θ=cos⁡θsin⁡θ\begin{array}{ccc}
\tan{\theta}=\displaystyle\frac{\sin\theta}{\cos{\theta}}&&&\cot{\theta}=\displaystyle\frac{\cos\theta}{\sin\theta}
\end{array}tanθ=cosθsinθ​​​​cotθ=sinθcosθ​​

Example

Express sin⁡3xcos⁡3x\displaystyle\frac{\sin3x}{\cos3x}cos3xsin3x​ as a single trigonometric function.

sin⁡3xcos⁡3x=tan⁡3x\displaystyle\frac{\sin3x}{\cos3x}=\tan3xcos3xsin3x​=tan3x

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Example: Reciprocal & Quotient Identities

Simplify csc⁡θ−sin⁡θ\csc\theta-\sin\thetacscθ−sinθinto one trigonometric ratio.

csc⁡θ−sin⁡θ=1sin⁡θ−sin⁡θReciprocal Identity=1−sin⁡2θsin⁡θ\begin{array}{rcl}
\csc\theta-\sin\theta&=&\displaystyle\frac{1}{\sin\theta}-\sin\theta&&\footnotesize{\color{orange}\text{Reciprocal Identity}}\\\\
&=&\displaystyle\frac{1-\sin^{2}\theta}{\sin\theta}
\end{array}cscθ−sinθ​==​sinθ1​−sinθsinθ1−sin2θ​​Reciprocal Identity

Practice: Reciprocal & Quotient Identities
Which of the following is equivalent to tan⁡θcsc⁡θ\displaystyle\frac{\tan\theta}{\csc\theta}cscθtanθ​?

A) sin⁡2θcos⁡θ\displaystyle\frac{\sin^{2}\theta}{\cos\theta}cosθsin2θ​ 

B) tan⁡θ\tan\thetatanθ 

C) sec⁡θ\sec\thetasecθ  

D) None of the above  

I don't know

Practice: Reciprocal & Quotient Identities
True or false: tan⁡θcos⁡2θsec⁡θ=sin⁡θ\displaystyle\frac{\tan\theta\cos^{2}\theta}{\sec\theta}=\sin\thetasecθtanθcos2θ​=sinθ.

Answer

I don't know

Practice: Reciprocal & Quotient Identities
Simplify tan⁡θcos⁡θ3sec⁡θcot⁡θ\displaystyle\frac{\tan\theta\cos\theta}{3\sec\theta\cot\theta}3secθcotθtanθcosθ​.

A) 13\displaystyle\frac{1}{3}31​ 

B) 13csc⁡2θ\displaystyle\frac{1}{3\csc^{2}\theta}3csc2θ1​ 

C) csc⁡2θ3\displaystyle\frac{\csc^{2}\theta}{3}3csc2θ​ 

D) sin⁡2θ3\displaystyle\frac{\sin^{2}\theta}{3}3sin2θ​ 

I don't know

Extra Practice

Reciprocal & Quotient Identities

Practice: Reciprocal & Quotient Identities
Simplify cot⁡θsin⁡θcsc⁡θtan⁡θ\dfrac{\cot\theta\sin\theta}{\csc\theta\tan\theta}cscθtanθcotθsinθ​.

Reciprocal & Quotient Identities

Practice: Reciprocal & Quotient Identities
Simplify cot⁡θ+12cos⁡θ\cot\theta+\dfrac{1}{2}\cos\thetacotθ+21​cosθ.

Reciprocal & Quotient Identities

Practice: Reciprocal & Quotient Identities
True or false: tan⁡θcsc⁡2θsec⁡θ=sec⁡θ\dfrac{\tan\theta\csc^{2}\theta}{\sec\theta}=\sec\thetasecθtanθcsc2θ​=secθ.

Reciprocal & Quotient Identities

Practice: Reciprocal & Quotient Identities
True or false: cot⁡2θsin⁡θcsc⁡θ=cos⁡2θ\dfrac{\cot^2\theta\sin\theta}{\csc\theta}=\cos^2\thetacscθcot2θsinθ​=cos2θ.

Reciprocal & Quotient Identities

Practice: Reciprocal & Quotient Identities
Which of the following is equivalent to cot⁡2θcos⁡θ\dfrac{\cot^2\theta}{\cos\theta}cosθcot2θ​?

Reciprocal & Quotient Identities

Practice: Reciprocal & Quotient Identities
Which of the following is equivalent to cot⁡θsec⁡θ\dfrac{\cot\theta}{\sec\theta}secθcotθ​?