Trig Segments
By Leighton McIntyre
Goal: To derive trigonometric ratios from given
segments
Problem
Solution
Circle O is a unit circle
Length OA = 1
The following ratios exist
AB =
AB
OA
= sinθ
OB =
OB
OA
= cosθ
CD
OD
=
AB
=
OB
OC
OA
=
1
=
cosθ
€
CD =
€
OE =
€
€
€
€
tanθ
secθ
OF =
OF
OA
=
1
OA =
1
cos(90 − θ )
=
1
=
sin θ
OF
GH =
€
€
AE =
€
AF =
€
= tan (90 - θ)€ = cotθ
€
AE
OA
= tanθ
AF
OA
= tan (90 - θ) = cotθ
OC
OD
=
OC =
€
OH =
€
GH
GO
€
OH
OG
OA
= 1 =
OB
cosθ
=
€
1
cos(90 − θ )
€
=
secθ
1
=
sin θ
€
cosecθ
€
Identities
from the diagram
€
€
€
1)
1
=
sin θ
2)
1
=
cosθ
secθ
3)
sin θ
=
cosθ
tanθ
cosecθ
Trig Identities derived
1) Sin θ=
1
cosecθ
2) Cos θ =
€
3) tan θ =
€
€
1
sec θ
1
cot θ
cosecθ