Trig – In a Nutshell

Help I’m trapped in a nutshell

(1,0)

(0,-1)

(0,1)

(-1,0)

+,+

-,-

-,+

+,-

The Unit Circle

+

-

Term

inal

ray

θ

2

2

32

These are the reference triangles

12 30 6

32

122

45 4

22

1

12

60 3

321

Trigonometric Functions

θ

opp y

sinhyp r

x

yr

sin opp y

tancos adj x

adj x

coshyp r

1 hyp rsec

cos adj x

1 cos adj x

cottan sin opp y

1 hyp rcsc

sin opp y

The Identities

2 2sin x cos x 1

2 21 tan x sec x

2 21 cot x csc x

Lets go graphing

p.s. – are cos x and sin x odd or even functions?

d?

Move graph c?

Moves graph a?

Changes Amplitudeb?

Changes Period 2coeff of x

by fun( x c)a d

Graphing Trig Functions (my method)

Sin/Cos

1. Determine New Start: Set Argument = 02. Determine New End: Set Argument = 2π (or New Start

+ Period)3. Find 3 midpoints

4. Plot these 5 points

Graphing Trig Functions (my method)

Tan/Cot/Sec/Csc

1. Determine 3 New Asymptotes Set Argument = Old Asymptotes and

solve for x2. Find 2 midpoints3. Plot these 2 points

4. Sketch the functions according to what they looked like before.

Angle Addition Formulas

sin A B sinAcosB cosAsinB

cos A B cosAcosB sinAsinB

cos A B cosAcosB sinAsinB

sin A B sinAcosB cosAsinB

tanA tanB

tan A B1 tanAtanB

tanA tanB

tan A B1 tanAtanB

2 2cos2A cos A sin A

sin2A 2sinAcosA

2

2tanAtan2A

1 tan A

Double/Half Angle

Double/Half Angle

A 1 cosAsin

2 2

A 1 cosA

cos2 2

A sinA 1 cosA

tan2 1 cosA sinA

2 2 2c a b 2abcosCLaw of Cosines

Inverse Trig

For sin-1x, csc-1x, tan-1xQI or QIV (go clockwise!)

For cos-1x, sec-1x, cot-1xQI or QII

y

2 2

0 y