Reciprocal identities
Define cosecant, secant, and cotangent as reciprocals.
Reciprocal identities is one of 5 identities formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why reciprocal identities works
Flipping a fraction over is the same as dividing 1 by it. Since sine is opposite over hypotenuse, one divided by sine is hypotenuse over opposite, and that ratio is what cosecant names. The three new functions add no new information about the triangle; they just save you from writing a fraction inside a fraction.
What each symbol means
$\theta$ is an angle.
Reciprocal identities: when it holds
The denominator function must be nonzero.
When it stops applying
Cotangent as 1 over tangent breaks at 90 degrees, because tan 90 degrees does not exist and you cannot divide 1 by nothing. Yet cot 90 degrees is a perfectly good 0. At the angles where tangent blows up, get cotangent from cos over sin instead.
Reciprocal identities: a worked example
If $\cos\theta=2/3$, then $\sec\theta=3/2$.
The mistake to avoid
What people do: Students read the −1 in sin⁻¹ as an exponent and hand in sin⁻¹(0.5) when the question asked for csc(0.5).
Why it goes wrong: On a calculator sin⁻¹ is the inverse sine key, which answers the question which angle has this sine. Cosecant answers a completely different question, and the two almost never give the same number.
Do this instead: Use the reciprocal key instead. Type sin of the angle, then press the 1/x button. For an angle of 30 degrees that gives csc 30 = 2, while sin⁻¹(30) is not even defined.
Reciprocal identities: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\theta$ is an angle.
- Check the conditions before substituting. The denominator function must be nonzero.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Identities
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where reciprocal identities comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about reciprocal identities
Why is secant paired with cosine when both words start differently?
The names come from old circle diagrams, not from the spelling. It helps to notice the pairing is crossed: the co- function cosecant belongs to plain sine, and plain secant belongs to cosine.
My calculator has no csc button, how do I get the value?
Compute the sine first, then press the reciprocal key or divide 1 by the result. If cos of the angle is 2/3, then sec is 3/2 = 1.5, and the same two keystrokes work for the other two.
Can cosecant equal zero?
Never. It is 1 divided by a sine, and a fraction with 1 on top can never come out as 0. In fact cosecant is always 1 or more in size, because sine never exceeds 1.
Where are these three functions actually undefined?
Wherever the function underneath hits zero. Cosecant fails at 0 and 180 degrees where sine is 0, secant fails at 90 and 270 degrees where cosine is 0, and cotangent fails at 0 and 180 degrees.