The flipped ratios
Turn a ratio upside down. It gets a new name.
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Step 1 of 4
- A right triangle: opposite, adjacent, hypotenuse.
- Sine flips to cosecant: hypotenuse over opposite.
- Cosine flips to secant: hypotenuse over adjacent.
- Tangent flips to cotangent: adjacent over opposite.
Three flips
A ratio's flip swaps its top and bottom. The flip of 4/5 is 5/4.
Cosecant, secant and cotangent are the flips of sine, cosine and tangent. No new side is measured.
Cosec is short for cosecant, sec for secant, and cot for cotangent.
Rule: cosec A = 1 ÷ sin A. sec A = 1 ÷ cos A. cot A = 1 ÷ tan A.
| Ratio | Flip | Product |
|---|---|---|
| sin A | cosec A | sin A × cosec A = 1 |
| cos A | sec A | cos A × sec A = 1 |
| tan A | cot A | tan A × cot A = 1 |
Take a 5-12-13 triangle where A faces the side 5. Here sin A = 5/13, so cosec A = 1 ÷ (5/13) = 13/5. Also cos A = 12/13, so sec A = 13/12. And tan A = 5/12, so cot A = 12/5.
Tangent and secant are also names of lines that meet a circle. Here they name ratios of sides, not lines.
Watch out!
Cosec is the flip of sin, not of cos. Sec is the flip of cos. Do not match them by first letters.