WHAT

I LE

ARNED FROM

CREATIN

G AN A

DVANCED

TRIG

CLA

SS

DR

. K

AT

I E C

ER

RO

NE

TH

E U

NI V

ER

SI T

Y O

F A

KR

ON

CO

LLE

GE

OF

AP

PLI E

D S

CI E

NC

E A

ND

TE

CH

NO

LOG

Y

BACKGROUNDTechnical College

Our programs

Accreditation

Professional Exams

Replaced Tech Calc II

Advanced Trig

Advanced Topics

THE ADVANCED TRIG COURSE1. Circles and Circular Curves : Arcs and central angles;

Chords and segments, Secant and tangent lines,, Perpendicular bisectors; Lengths of tangent lines, chords, curves, external distances and middle ordinates; Circular curve computation

2. Parabolic Curves: Slope of a line (grade or gradient); Distance of a line; Points of vertical curvature, intersection, and tangency; Tangent elevations; Basic form of a parabola; Finding the external distance of a vertical curve

3. Spherical Trigonometry: Spherical triangles, Interior and dihedral angles; Sine formulas for spherical triangles; Cosine formulas for sides of spherical triangles; Cosine formulas for angles of spherical triangles; Applications of spherical triangles

PARABOLIC CURVESGiven: focal length f

PARABOLIC CURVES

• Point of Curvature (PC): the beginning of the arc

• Point of Intersection (PI): The point where the two tangents intersect

• Point of Tangency (PT):The end of the arc

• Length of the Chord (L): The length from PC to PT

PTPC

PI

L

𝑔1 𝑔2

PARABOLIC CURVESGiven:

PTPC

PI

L

𝑔1 𝑔2

PARABOLIC CURVESGiven:

PTPC

PI

L

𝑔1

PARABOLIC CURVESGiven:

Let x = 0

PTPC

PI

L

𝑔1

PARABOLIC CURVESA -1.500% grade meets a +2.250% grade at station 36+50 (3650 ft) , elevation 452.00 ft. A vertical curve of length 600 ft. (6 stations) will be used.

PTPC

PI = 452 ft.

L = 6

-1.5 2.25

PARABOLIC CURVESTURNING POINT

PTPC

PI = 452 ft.

L

PARABOLIC CURVESTURNING POINTA -1.500% grade meets a +2.250% grade at station 36+50 (3650 ft) , elevation 452.00 ft. A vertical curve of length 600 ft. (6 stations) will be used.

PTPC

PI = 452 ft.

L = 6

-1.5 2.25

OR

CIRCULAR CURVES

• Point of Curvature (PC): the beginning of the arc

• Point of Intersection (PI): The point where the two tangents intersect

• Point of Tangency (PT):The end of the arc

• Length of the long chord (L): The length from PC to PT

PTPC

PI

L

CIRCULAR CURVES

• Tangent distance (T): The distance from PI to PC or from PI to PT

• Deflection Angle(Δ): The central angle of the angle at the Point of Intersection (PI)

PTPC

PI

L

T T

RR

• Length of the Curve (C): the arclength from PC to PT• Radius (R): Radius of the circle• Degree of a Curve (D): the central angle that subtends a 100

foot arc

CIRCULAR CURVESGiven D and Δ, find R. PTPC

PI

L

T T

RR

CIRCULAR CURVESLength of the Curve (L): The arclength from PC to PTGiven R and Δ

PTPC

Δ/2L/2

R

CIRCULAR CURVESTangent distance (T): The distance from PI to PC or from PI to PTGiven R and Δ, find T.

PTPC

PI

T T

RR

Δ/2

CIRCULAR CURVESGiven D and Δ, find C. PTPC

PI

L

T T

RR

CIRCULAR CURVES• External Distance (E): The

distance from the Point of Intersection to the middle of the curve

• Middle Ordinate (M): the length of the ordinate from the middle of the long chord to the middle of the arc

PTPC

PI

L

T T

RR

E

M

CIRCULAR CURVESGiven R and Δ, find E. PTPC

PI

L/2

T T

RR

E

Δ/2

CIRCULAR CURVESGiven D and Δ

PTPC

PI

L/2

T T

RR

M

Δ/2𝑎

INCREASED TEST SCORES

KATIE

CER

RONE

kc24@uakron.edu

The

Univer

sity

of A

kron

Colle

ge of

Applie

d Sci

ence

and T

echnol

ogy