PreCalc 2 · Analytic Trigonometry
Find the exact value of an inverse trig function, or of a trig function of an inverse. Stay inside the range of the inverse.
Ranges
arcsin x is from −π2 to π2.
arccos x is from 0 to π.
arctan x is from −π2 to π2.
A positive value lands in Quadrant I. A negative arcsin or arctan lands in Quadrant IV. A negative arccos lands in Quadrant II.
arcsin x is also written sin⁻¹ x. arccos x is also written cos⁻¹ x. arctan x is also written tan⁻¹ x.
One inverse
Read the unit-circle angle that sits inside the range.
arcsin 12 = π6
arccos(−12) = 2π3
A triangle
For sin(arccos x), draw a right triangle whose cosine is x.
The hypotenuse is positive.
The quadrant from the range tells you each sign.
Inverse of a trig value
sin(arcsin x) = x only when x is already in the arcsin range.
arcsin(sin 5π6) = π6, because 5π6 is outside the range and sine is still 12.
One inverse of a unit-circle value.
arcsin x is the angle from −π2 to π2 whose sine is x.
arcsin 2
π4
Answer: π4
arcsin x is the angle from −π2 to π2 whose sine is x.
arcsin 12
π6
Answer: π6
arcsin x is the angle from −π2 to π2 whose sine is x.
arcsin 2
π3
Answer: π3
arcsin x is the angle from −π2 to π2 whose sine is x.
arcsin −2
−π4
Answer: −π4
arcsin x is the angle from −π2 to π2 whose sine is x.
arcsin −1
−π2
Answer: −π2
arccos x is the angle from 0 to π whose cosine is x.
arccos 12
π3
Answer: π3
arccos x is the angle from 0 to π whose cosine is x.
arccos 0
π2
Answer: π2
arccos x is the angle from 0 to π whose cosine is x.
arccos −12
2π3
Answer: 2π3
arccos x is the angle from 0 to π whose cosine is x.
arccos −1
π
Answer: π
arctan x is the angle from −π2 to π2 whose tangent is x.
arctan
π3
Answer: π3
arctan x is the angle from −π2 to π2 whose tangent is x.
arctan −1
−π4
Answer: −π4
A trig function of an inverse, or an inverse of a trig function.
Let θ = arccos 45.
cos θ = 45
arccos returns an angle from 0 to π.
The cosine is positive, so θ is in Quadrant I.
opp² + adj² = hyp²
opp² + 4² = 5²
opp² = 25 − 16
opp² = 9
opp = 3
Sine is positive in this quadrant.
sin θ = opphyp
sin θ = 35
Answer: 35
Let θ = arccos 35.
cos θ = 35
arccos returns an angle from 0 to π.
The cosine is positive, so θ is in Quadrant I.
opp² + adj² = hyp²
opp² + 3² = 5²
opp² = 25 − 9
opp² = 16
opp = 4
Sine is positive in this quadrant.
sin θ = opphyp
sin θ = 45
Answer: 45
Let θ = arcsin 513.
sin θ = 513
arcsin returns an angle from −π2 to π2.
The sine is positive, so θ is in Quadrant I.
opp² + adj² = hyp²
5² + adj² = 13²
adj² = 169 − 25
adj² = 144
adj = 12
Cosine is positive in this quadrant.
cos θ = adjhyp
cos θ = 1213
Answer: 1213
Let θ = arcsin 817.
sin θ = 817
arcsin returns an angle from −π2 to π2.
The sine is positive, so θ is in Quadrant I.
opp² + adj² = hyp²
8² + adj² = 17²
adj² = 289 − 64
adj² = 225
adj = 15
tan θ = oppadj
tan θ = 815
Answer: 815
Let θ = arctan 2.
tan θ = 2
arctan returns an angle from −π2 to π2.
The tangent is positive, so θ is in Quadrant I.
Opposite 2, adjacent 1.
hyp² = opp² + adj²
hyp² = 4 + 1
hyp² = 5
hyp =
sin θ = opphyp
sin θ = 2
sin θ = 25
25
Answer: 25
Let θ = arcsin −725.
sin θ = −725
arcsin returns an angle from −π2 to π2.
The sine is negative, so θ is in Quadrant IV.
opp² + adj² = hyp²
7² + adj² = 25²
adj² = 625 − 49
adj² = 576
adj = 24
Cosine is positive in this quadrant.
cos θ = adjhyp
cos θ = 2425
Answer: 2425
tan 5π6 = −3
arctan returns an angle from −π2 to π2.
arctan −3
−π6
Answer: −π6
Let θ = arctan −34.
tan θ = −34
arctan returns an angle from −π2 to π2.
The tangent is negative, so θ is in Quadrant IV.
Opposite 3, adjacent 4.
hyp² = opp² + adj²
hyp² = 3² + 4²
hyp² = 25
hyp = 5
sin θ = opphyp
sin θ = −35
sin θ = −35
Answer: −35
sin 5π6 = 12
arcsin returns an angle from −π2 to π2.
arcsin 12
π6
Answer: π6
cos 4π3 = −12
arccos returns an angle from 0 to π.
arccos −12
2π3
Answer: 2π3
Let α = arcsin 35.
sin α = 35
arcsin returns an angle from −π2 to π2.
α is in Quadrant I.
cos α = 45
Let β = arccos 513.
cos β = 513
arccos returns an angle from 0 to π.
β is in Quadrant I.
sin β = 1213
cos(α + β) = cos α cos β − sin α sin β
cos α · cos β = 45 · 513
413
413
sin α · sin β = 35 · 1213
3665
413 − 3665
413 − 3665
−1665
Answer: −1665
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