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Wize High School Grade 11 Math Textbook > Trigonometry

Reciprocal Trig Ratios

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Reciprocal Trigonometric Ratios -- csc, sec, cot

The reciprocal trig ratios are:
csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc \theta = \dfrac{1}{\sin \theta},\quad \sec \theta = \dfrac{1}{\cos \theta},\quad \cot \theta = \dfrac{1}{\tan \theta}
We can also think of them as swapping the order of the primary trig ratios!

sin⁡θ=OppHypcos⁡θ=AdjHyptan⁡θ=OppAdjcsc⁡θ=HypOppsec⁡θ=HypAdjcot⁡θ=AdjOpp\large \boxed{ \begin{array}{c|c|c} \sin\theta=\dfrac{Opp}{Hyp} \quad& \cos\theta=\dfrac{Adj}{Hyp} \quad& \tan\theta=\dfrac{Opp}{Adj}\\[2em] \csc\theta=\dfrac{Hyp}{Opp} \quad& \sec\theta=\dfrac{Hyp}{Adj} \quad& \cot\theta=\dfrac{Adj}{Opp}\\ \end{array} }

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Calculator Time!

My calculator doesn't have reciprocal trig ratios!
That's ok, just follow these examples to find the answers using primary trig ratios:
csc⁡30°=1sin⁡30°sec⁡30°=1cos⁡30°cot⁡30°=1tan⁡30°\csc 30\degree = \dfrac{1}{\sin30\degree}\qquad \sec 30\degree = \dfrac{1}{\cos30\degree} \qquad \cot 30\degree = \dfrac{1}{\tan30\degree}
Watch Out!
Make sure your calculator is in the correct degree (D) mode.

Use your calculator to evaluate the following trig ratios:

1.) csc⁡60°\csc60\degree
csc⁡60°=1sin⁡(60°)≈10.866≈1.15\csc60\degree= \dfrac{1}{\sin(60\degree)}\approx\dfrac{1}{0.866} \approx \boxed{1.15}

2.) sec⁡ 90°\sec\ 90\degree
sec⁡90°=1cos⁡(90°)=10  ⟹  undefined (error)\sec90\degree= \dfrac{1}{\cos(90\degree)}=\dfrac{1}{0} \implies \boxed{\text{undefined (error)}}

3.) cot⁡45°\cot45\degree
cot⁡45°=1tan⁡(45°)=11=1\cot45\degree= \dfrac{1}{\tan(45\degree)}=\dfrac{1}{1} = \boxed{1}

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Practice: Primary Trig Ratios

Given the following triangle, find all of the reciprocal trig ratios.


csc⁡θ=HypOpp=1312sec⁡θ=HypAdj=135cot⁡θ=AdjOpp=512\begin{array}{ccccc} \begin{array}{rcl} \csc\theta&=&\dfrac{Hyp}{Opp}\\[1em] &=&\dfrac{13}{12} \end{array} && \begin{array}{|rclc|} \quad\sec\theta&=&\dfrac{Hyp}{Adj} &\\[1em] &=&\dfrac{13}{5} \end{array} && \begin{array}{rcl} \cot\theta&=&\dfrac{Adj}{Opp}\\[1em] &=&\dfrac{5}{12} \end{array} \end{array}

Practice: Primary Trig Ratios

Use a calculator to evaluate the following.

a) sec⁡60°=\sec60\degree=

b) csc⁡20°=\csc20\degree=

c) cot⁡10°=\cot10\degree=

*Round your answer to 2 decimal places. Enter DNE if the answer is undefined (error).

Practice: Primary Trig Ratios

Given the following triangle, find all of the reciprocal trig ratios.

Practice: Primary Trig Ratios

A family rents a truck to move to a new house. The truck has a ramp that extends down from the back.
The truck is backed up to be exactly 6 meters from the door of the house, and the ramp makes an angle of 10°10\degree with the ground.

Use a reciprocal trig ratio to determine the length of the ramp.