Secant
Reciprocal of cosine.
Formula
Variables
Example
sec 60° = 2
How it works
The secant of an angle is defined as the reciprocal of its cosine, so it tells you "1 divided by cosine" for that angle. This comes directly from the right-triangle and unit-circle definitions of cosine: wherever cosine gives the horizontal ratio, secant gives the multiplicative inverse of that ratio. Because of that, secant becomes large when cosine is small, and it is undefined whenever cosine is 0.
When to use it
Use secant when a problem gives you cosine and asks for the reciprocal value, or when a trigonometric expression is written in terms of secants. It is also useful in geometry, physics, and calculus when angles are involved and you need to rewrite everything in a consistent trig form.
Worked example, step by step
Find sec 45°. cos 45° = √2/2 sec 45° = 1 / cos 45° sec 45° = 1 / (√2/2) sec 45° = 2/√2 sec 45° = √2 Final answer: sec 45° = √2
Common mistakes to avoid
- Using secant as if it means 1/sine instead of 1/cosine
- Forgetting that secant is undefined when cos θ = 0
- Leaving the answer as 1/(√2/2) instead of simplifying it
Did You Know?
Secant comes from the Latin for "cutting", because the line it measures cuts across the circle.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Cotangent
IntermediateReciprocal of tangent.
Tangent Identity
BasicTangent equals sine divided by cosine.