Inverse Reciprocal Trig: Step-by-Step Guide
The inverse reciprocal trig functions arcsec(x), arccsc(x), and arccot(x) undo secant, cosecant, and cotangent. You can find them by flipping x and using a regular inverse: arcsec(x) = arccos(1/x), arccsc(x) = arcsin(1/x), and arccot(x) = arctan(1/x) when x is positive (it needs an adjustment for negative x).
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Figure out which function it is. Is it arcsec, arccsc, or arccot?\[y = \operatorname{arcsec}(x),\;\operatorname{arccsc}(x),\;\text{or}\;\operatorname{arccot}(x)\]
- Check the input. arcsec and arccsc need |x| to be at least 1. arccot works for any real number.\[|x| \geq 1\;\text{(for arcsec, arccsc)}\]
- Rewrite as a regular inverse. Flip x (use 1/x) and use arccos, arcsin, or arctan.\[\operatorname{arcsec}(x) = \arccos\!\left(\tfrac{1}{x}\right),\; \operatorname{arccsc}(x) = \arcsin\!\left(\tfrac{1}{x}\right),\; \operatorname{arccot}(x) = \arctan\!\left(\tfrac{1}{x}\right) \text{ (x>0)}\]
- Find the angle. Use the unit circle values you have learned.\[\theta = \arccos\!\left(\tfrac{1}{x}\right) \in [0, \pi]\]
- Adjust for the range. For arccot(x) when x is negative, add pi so the answer is between 0 and pi. Check that the answer is in the right range.\[\operatorname{arccot}(x) = \arctan\!\left(\tfrac{1}{x}\right) + \pi \text{ if } x < 0\]
Worked example
Find arcsec(2).
Step 1: This is arcsec.
Step 2: The size of 2 is at least 1, so it is a valid input.
Step 3: arcsec(2) = arccos(1/2).
Step 4: arccos(1/2) = pi/3.
Step 5: pi/3 is between 0 and pi/2, which is the normal range for arcsec when x is positive.
The answer is arcsec(2) = pi/3.
Check: sec(pi/3) = 1 / cos(pi/3) = 1 / (1/2) = 2.
Tips and common mistakes
- arcsec(x) = arccos(1/x). This is the easiest one to remember.
- arccsc(x) = arcsin(1/x).
- arccot(x) = arctan(1/x) when x is positive. When x is negative, add pi: arccot(x) = arctan(1/x) + pi.
- arcsec and arccsc need |x| to be at least 1.
- Some textbooks use different ranges for arcsec and arccsc. Check which one your class uses.
- The range of arccot is between 0 and pi, not between -pi/2 and pi/2.
- It is often easiest to flip the number first and use a regular inverse function.
- On a calculator, find 1/x first, then press the regular inverse button.