Trigonometric identities and equations

Trigonometry laws and identities

trigonometric identities

Trig Table of Common Angles

Trig Table of Common Angles

Angle (degrees)

0

30

45

60

90

120

135

150

180

210

225

240

270

300

315

330

360 = 0

Angle (radians)

0

PI/6

PI/4

PI/3

PI/2

2/3PI

3/4PI

5/6PI

PI

7/6PI

5/4PI

4/3PI

3/2PI

5/3PI

7/4PI

11/6PI

2PI = 0 Square root of all given below terms

Sin(a)

(0/4)

(1/4)

(2/4)

(3/4)

(4/4)

(3/4)

(2/4)

(1/4)

(0/4)

-(1/4)

-(2/4)

-(3/4)

-(4/4)

-(3/4)

-(2/4)

-(1/4)

(0/4)

Cos(a)

(4/4)

(3/4)

(2/4)

(1/4)

(0/4)

-(1/4)

-(2/4)

-(3/4)

-(4/4)

-(3/4)

-(2/4)

-(1/4)

(0/4)

(1/4)

(2/4)

(3/4)

(4/4)

Tan(a)

(0/4)

(1/3)

(2/2)

(3/1)

(4/0)

-(3/1)

-(2/2)

-(1/3)

-(0/4)

(1/3)

(2/2)

(3/1)

(4/0)

-(3/1)

-(2/2)

-(1/3)

(0/4)

Those with a zero in the denominator are undefined (Not Defined) . They are included solely to demonstrate the pattern.

Basic Trigonometric Identities

Pythagorean Identities (and their variations)

From the unit circle, using the Pythagorean Theorem, we see that

x² + y² = 1

Since x = cos θ & y = sin θ, we get:

(sin θ)² + (cos θ)² = 1 or

sin² θ + cos² θ = 1

1. sin2 θ + cos2 θ = 1 (sin2 θ = 1 - cos2 θ) (cos2 θ = 1 - sin2 θ)

2. tan2 θ + 1 = sec2 θ (tan2 θ - sec2 θ = 1) (tan2 θ = sec2 θ – 1)

3. 1 + cot2 θ = csc2 θ (cot2 θ =  csc2 θ – 1) (1 =  csc2 θ - cot2 θ)

Opposite Angle Identities (aka Even/Odd Identities)

1. sin (-A) = - sin (A)                2. cos (-A) = cos (A)

Sum and Difference Identities

1.    sin(A + B) = sin(A)cos(B) + sin(B)cos(A)

2.  sin(A - B) = sin(A)cos(B) - sin(B)cos(A)

3.  cos(A + B) = cos(A)cos(B) – sin(A)sin(B)

4.  cos(A – B) = cos(A)cos(B) + sin(A)sin(B)

Cofunction Identities:  

1. sin θ = cos (90° – θ)	4. cos θ = sin (90° – θ)
2. tan θ = cot (90° – θ)	5. cot θ = tan (90° – θ)
3. sec θ = csc (90° – θ)	6. csc θ = sec (90° – θ)