01 · Standard angles
Find sec(π/4), cosec(π/3) and cot(π/6).
Hint
Start from the exact sine and cosine values.
Worked solution
√2, 2√3/3 and √3 respectively. For example, cosec(π/3) = 1/(√3/2) = 2√3/3.
Understand · explore · practise
Sec, cosec and cot
Learn sec, cosec and cot with exact values, quadrant signs and domain restrictions. Explore a unit circle, see the tangent geometry and practise with worked solutions.
Before you startSine, cosine, tangent, radians and exact special-angle values
01 / Three ratios built from familiar ones
Start with sine and cosine, then check the denominator.
Secant (sec), cosecant (cosec, also written csc) and cotangent (cot) are trigonometric functions. You can calculate them from the sine and cosine of the same angle.
sec θ = 1 / cos θ, when cos θ ≠ 0
cosec θ = 1 / sin θ, when sin θ ≠ 0
cot θ = cos θ / sin θ, when sin θ ≠ 0
Choose an angle. A negative coordinate gives a negative reciprocal. At a zero denominator the expression is undefined; it does not equal infinity.
Coordinates and their reciprocalsExplore
θ = 5π/6 rad.
sec θ = −2√3/3; cosec θ = 2; cot θ = −√3.
All three values are defined.
Blue: sec. Green: cosec. Gold: cot. An undefined value has no dot. The number lines show values, not lengths.
02 / A reciprocal is not an inverse function
Taking one over a value is different from finding an angle.
Compare three operationsWorked example
cos(π/3) = 1/2
The input is an angle and the output is a ratio.
sec(π/3) = 1/(1/2) = 2
Take the reciprocal of that ratio.
arccos(1/2) = π/3
The inverse cosine function returns a principal angle.
On a calculator, the key labelled cos⁻¹ usually means arccos, not sec. To calculate sec θ, enter 1 ÷ cos(θ), using brackets. Likewise use 1 ÷ sin(θ) for cosec and cos(θ) ÷ sin(θ) for cot.
(cos θ)⁻¹ = 1/cos θ = sec θ
cos⁻¹(x), as inverse-function notation, means arccos(x)
The placement of the exponent and the context matter. Writing arccos avoids the ambiguity.
03 / Find exact values before rounding
Invert the entire fraction, then simplify.
Calculate sec(π/6), cosec(π/4) and cot(π/3)Worked example
sec(π/6) = 1/(√3/2) = 2/√3 = 2√3/3
A reciprocal swaps numerator and denominator.
cosec(π/4) = 1/(√2/2) = 2/√2 = √2
Rationalise and simplify.
cot(π/3) = (1/2)/(√3/2) = 1/√3 = √3/3
Use cosine divided by sine.
For θ = π/6, π/4, π/3, the sec values are 2√3/3, √2, 2; the cosec values are 2, √2, 2√3/3; and the cot values are √3, 1, √3/3.
04 / Keep the quadrant signs
Reciprocals preserve signs; quotients compare them.
Sec has the sign of cosine. Cosec has the sign of sine. Cot is positive when sine and cosine have the same sign, and negative when their signs differ.
Quadrant I: all three positive
Quadrant II: sec −, cosec +, cot −
Quadrant III: sec −, cosec −, cot +
Quadrant IV: sec +, cosec −, cot −
Find sec(5π/6), cosec(7π/6) and cot(7π/4)Worked example
cos(5π/6) = −√3/2, so sec(5π/6) = −2√3/3
The reciprocal keeps the negative sign.
sin(7π/6) = −1/2, so cosec(7π/6) = −2
Quadrant III makes sine negative.
cot(7π/4) = (√2/2)/(−√2/2) = −1
The coordinate signs differ.
05 / Decide where each function exists
Check sine or cosine, rather than relying on a calculator error.
sec θ is undefined at θ = π/2 + nπ.
cosec θ and cot θ are undefined at θ = nπ.
Here n is any integer; angles are in radians.
At θ = 0, sec θ = 1 but cosec and cot are undefined. At θ = π/2, sec is undefined, cosec θ = 1 and cot θ = 0.
Why cot(π/2) exists even though tan(π/2) does notWorked example
cot(π/2) = cos(π/2)/sin(π/2) = 0/1 = 0
Its denominator is nonzero.
1/tan θ equals cot θ only where tan θ exists and is nonzero
The shortcut has a smaller domain.
At odd multiples of π/2, use cos θ/sin θ directly
Do not turn a valid cot value into an error by using an invalid intermediate expression.
Sec and cosec can never equal 0: their numerator is 1. Cot does equal 0 at θ = π/2 + nπ. For real inputs, |sec θ| ≥ 1 and |cosec θ| ≥ 1 wherever defined, since |cos θ| and |sin θ| are at most 1.
06 / Set the angle mode and keep brackets
The same written number describes different angles in degrees and radians.
Calculate sec(0.8) and cosec(40°), to 4 significant figuresWorked example
For sec(0.8), use RAD mode: 1 ÷ cos(0.8)
No degree symbol here: the angle is in radians.
sec(0.8) ≈ 1.435
Keep the unrounded value if it feeds another calculation.
For cosec(40°), use DEG mode: 1 ÷ sin(40)
The degree symbol specifies the mode.
cosec(40°) ≈ 1.556
Do not press sin⁻¹: that would request an angle instead.
If a familiar exact angle should make the denominator 0, use the exact mathematics. A tiny floating-point denominator may produce a huge calculator number, but the exact function remains undefined.
07 / See the ratios as tangent lengths
For an acute angle, one tangent creates three useful right triangles.
Let OP be a unit radius making an acute angle θ with the positive horizontal axis. The tangent at P meets the positive x-axis at X and y-axis at Y. A tangent is perpendicular to the radius at the point of contact.
Read the lengths from right trianglesWorked example
In OPX, cos θ = OP/OX = 1/OX, so OX = sec θ
OX is the hypotenuse of this triangle.
In OPY, ∠POY = π/2 − θ, so cos(π/2 − θ) = 1/OY
Hence sin θ = 1/OY, giving OY = cosec θ.
In OPY, PY/OP = tan(π/2 − θ) = cot θ
Since OP = 1, the tangent segment PY has length cot θ.
At θ = π/6, OX = 2√3/3, OY = 2 and PY = √3. This length interpretation uses an acute angle. Elsewhere, signed function values are not automatically positive lengths.
Watch the tangent intercepts change
Pause, replay or seek freely. The notes explain the same idea and stay in view.
08 / Recover other ratios from one value
A quadrant removes the sign ambiguity.
Given sec θ = −13/5 and π/2 < θ < π, find cosec θ and cot θWorked example
cos θ = −5/13
Invert the given value.
sin²θ = 1 −25/169 = 144/169
Use sin²θ + cos²θ = 1.
sin θ = 12/13
Quadrant II selects the positive root.
cosec θ = 13/12; cot θ = (−5/13)/(12/13) = −5/12
Check the signs against the quadrant.
Without the quadrant, sin θ could have either sign. Do not silently choose the positive root just because the square is positive.
09 / Reduce angles and handle powers
The reciprocal uses the value at the whole angle.
Keep the input and the operation separateWorked example
sec(−π/3) = 2; cosec(13π/6) = 2
Cosine is even, and sine repeats after 2π.
cot(5π/4) = 1
Both coordinates are negative and equal.
sec²(π/6) = (2√3/3)² = 4/3
sec²θ means (sec θ)², not sec(θ²).
2sec(π/3) − cosec(π/6) = 2(2) − 2 = 2
Evaluate the trigonometric functions before combining terms.
Sec is even and repeats after 2π. Cosec is odd and repeats after 2π. Cot is odd and repeats after π, on their respective domains.
10 / Your turn
Use exact values unless an accuracy is specified.
02 · Negative values
Find sec(2π/3), cosec(5π/4) and cot(5π/6).
Hint
Use the quadrant before taking a reciprocal.
Worked solution
−2, −√2 and −√3 respectively. Cot(5π/6) = (−√3/2)/(1/2) = −√3.
03 · At an axis
Find sec π, cosec(3π/2) and cot(3π/2).
Hint
The circle coordinates at 3π/2 are (0,−1).
Worked solution
−1, −1 and 0. In particular, cot(3π/2) = 0/(−1) = 0.
04 · Undefined inputs
Which are undefined: sec(3π/2), cosec π, cot 0, sec 0?
Hint
Check the denominator of each definition.
Worked solution
The first three are undefined; sec 0 = 1. Neither division by 0 nor infinity is a valid real function value.
05 · Inverse or reciprocal?
A student uses cos⁻¹(1/2) to calculate sec(π/3). Explain the error and give both correct results in radians.
Hint
One operation returns an angle; the other a ratio.
Worked solution
cos⁻¹(1/2) = π/3 means inverse cosine. Sec(π/3) = 1/(1/2) = 2.
06 · Extra turns
Find sec(−5π/3), cosec(17π/6) and cot(9π/4).
Hint
Add or subtract complete periods.
Worked solution
Sec(−5π/3) = sec(π/3) = 2; cosec(17π/6) = cosec(5π/6) = 2; cot(9π/4) = cot(π/4) = 1.
07 · Powers
Evaluate cosec²(π/6) − cot²(π/6) and sec(π/4)cosec(π/4).
Hint
Square or multiply the exact function values.
Worked solution
2² − (√3)² = 1; √2 × √2 = 2. The squares are applied after evaluating each ratio.
08 · Signed triangle
Cosec θ = −17/8 and π < θ < 3π/2. Find sec θ and cot θ.
Hint
Sin θ = −8/17; cosine is also negative.
Worked solution
Cos²θ = 1 −64/289 = 225/289, so cos θ = −15/17. Hence sec θ = −17/15 and cot θ = 15/8.
09 · Tangent geometry
In the unit-circle tangent diagram, θ = π/4. Find OX, OY and PY.
Hint
Use sec, cosec and cot respectively.
Worked solution
OX = √2, OY = √2, PY = 1. Each small right triangle has legs 1 and 1 and hypotenuse √2.
10 · Calculator
Find cot(0.6), to 4 significant figures.
Hint
Use RAD mode and cos(0.6)/sin(0.6).
Worked solution
Cot(0.6) = 1.461695… ≈ 1.462. An inverse tangent key performs a different operation.
11 · Possible values
Can sec θ = 0.4, cosec θ = −1.2 or cot θ = 0 for a real angle?
Hint
Check the ranges and axis values.
Worked solution
Sec θ = 0.4 is impossible: it would require cos θ = 2.5. Cosec θ = −1.2 is possible because sin θ = −5/6 is within [−1,1]. Cot θ = 0 is possible, for example θ = π/2.
12 · A domain trap
A calculator reports an error for 1/tan(π/2). Does that prove cot(π/2) is undefined?
Hint
Use the defining ratio cos θ/sin θ.
Worked solution
No. Tan(π/2) is undefined, so that shortcut cannot be used. Cot(π/2) = 0/1 = 0 is defined.
11 / Recap
Use the definitions, preserve the signs and check the denominator.
- Sec = 1/cos and cosec = 1/sin.
- Cot = cos/sin, including cot = 0 where cosine is 0.
- The shortcut cot = 1/tan needs tan to exist and be nonzero.
- Reciprocal functions are different from inverse functions.
- Special triangles give exact values; quadrant signs survive taking reciprocals.
- Sec and cosec never equal 0 and have magnitude at least 1.
Section 1 of 11 · Three ratios built from familiar ones