Every ratio has a reciprocal, found by flipping the fraction. Those three reciprocals get names of their own.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The three names
Cosecant is 1 ÷ sine. Secant is 1 ÷ cosine. Cotangent is 1 ÷ tangent.
The pairing looks wrong on purpose
Secant pairs with cosine, and cosecant pairs with sine. The prefixes cross over, which is the mistake almost everyone makes once.
Where they break
A reciprocal is undefined wherever the original ratio is zero. Since sin 0° = 0, csc 0° does not exist.
Why bother
They shorten formulas. The derivative of tangent is sec² x, which is far tidier than writing 1 ÷ cos² x every time.
Three reciprocals
Cosecant is 1/sine, secant is 1/cosine, cotangent is 1/tangent. They introduce no new information — every problem can be done without them — but they make many identities and integrals far shorter to write.
The pairing is not alphabetical
Secant pairs with cosine and cosecant with sine, which is the reverse of what the names suggest. Remembering that the co- prefixes cross over is the standard way people keep this straight.
Most calculators lack the keys
Compute the base ratio and take its reciprocal. There is no sec button on most machines because there is no need for one, which is itself a reminder that these are conveniences rather than new functions.
They are undefined where the base is zero
Cosecant is undefined wherever sine is zero, and secant wherever cosine is zero. Those points become vertical asymptotes on the graphs, and they are excluded from the domain of any identity involving them.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Priya Find a Secant
In a right triangle the adjacent leg is 3 and the hypotenuse is 5.
- Step 1
She writes the cosine first: cos θ = adjacent ÷ hypotenuse = 3/5.
Step 4: Your Turn
Practice makes it stick.
The Flip
Problem 1 of 2
sin θ = 0.5. What is csc θ?
The Pairs
Problem 2 of 2
Match each ratio with its reciprocal.
Tap a card on the left to start.
Flip the Ratio
1 of 8
cos θ = 0.25. What is sec θ?
2 of 8
tan θ = 4. What is cot θ, as a decimal?
3 of 8
sin θ = 0.2. What is csc θ?
4 of 8
Opposite 3, hypotenuse 5. What is csc θ, as a decimal to two places?
5 of 8
What is sec 0°?
6 of 8
cot θ = 1. What is tan θ?
7 of 8
Which of these are undefined?
8 of 8
Adjacent 8, opposite 6. What is cot θ, as a decimal to two places?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Which ratio is the reciprocal of cosine?
Question 2 of 2
cos θ = 0.5. What is sec θ?
What You Learned
- Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent.
- The prefixes cross over: secant pairs with cosine.
- A reciprocal is undefined wherever the original ratio is zero.