TRIGONOMETRY LAWS AND IDENTITIES - California State University San Marcos

TRIGONOMETRY

QUOTIENT IDENTITIES

tan(x)

=

sin(x) cos(x)

cot(x)

=

cos(x) sin(x)

RECIPROCAL IDENTITIES

csc(x) = 1 sin(x)

sec(x) = 1 cos(x)

cot(x)

=

1 tan(x)

sin(x) = 1 csc(x)

cos(x) = 1 sec(x)

tan(x)

=

1 cot(x)

EVEN/ODD IDENTITIES

sin( x) = sin(x) cos( x) = cos(x) tan( x) = tan(x)

csc( x) = csc(x) sec( x) = sec(x) cot( x) = cot(x)

PYTHAGOREAN IDENTITIES

cos2(x) + sin2(x) = 1 tan2(x) + 1 = sec2(x) cot2(x) + 1 = csc2(x)

SUM IDENTITIES

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

cos(x + y) = cos(x) cos(y) sin(x) sin(y)

tan(x

+

y)

=

tan(x) + tan(y) 1 tan(x) tan(y)

DIFFERENCE IDENTITIES

sin(x cos(x

tan(x

y) = sin(x) cos(y) cos(x) sin(y)

y) = cos(x) cos(y) + sin(x) sin(y)

y)

=

tan(x) tan(y) 1 + tan(x) tan(y)

LAW OF SINES

sin(A) a

=

sin(B) b

=

sin(C ) c

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LAWS AND IDENTITIES

DOUBLE-ANGLE IDENTITIES

sin(2x) = 2 sin(x) cos(x)

cos(2x) = cos2(x) sin2(x)

= 2 cos2(x) 1

= 1 2 sin2(x)

tan(2x)

=

1

2 tan(x) tan2(x)

HALF-ANGLE IDENTITIES

x

r 1 cos(x)

sin

=?

cos

2 x

=

r

?

1

+

2 cos(x)

2

s2

x

1 cos(x)

tan

=?

2

1 + cos(x)

PRODUCT TO SUM IDENTITIES

sin(x) sin(y)

=

1 2

[cos(x

y)

cos(x + y)]

cos(x) cos(y) = 1 [cos(x + y) + cos(x y)] 2

sin(x) cos(y) = 1 [sin(x + y) + sin(x y)] 2

cos(x) sin(y)

=

1 2

[sin(x

+

y)

sin(x

y)]

SUM TO PRODUCT IDENTITIES

sin(x) + sin(y) = 2 sin x + y cos x y

sin(x) cos(x)

+

sin(y) cos(y)

= =

2 2

cos cos

x x

2 +y 2 +y 2 x+

y scionsxxx222

y y

y

cos(x) cos(y) = 2 sin 2 sin 2

LAW OF COSINES

a2 = b2 + c2 b2 = a2 + c2 c2 = a2 + b2

2bc cos(A) 2ac cos(B) 2ab cos(C)

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Tel: STEM SC (N): (760) 750-4101 STEM SC (S): (760) 750-7324

TRIGONOMETRY

DEFINITIONS

sin(x) = Opposite csc(x) = Hypotenuse

H ypotenuse

Opposite

cos(x) = Adjacent sec(x) = Hypotenuse

H ypotenuse

Adjacent

tan(x) = Opposite Adjacent

cot(x) = Adjacent Opposite

Opposite

Hypotenuse

x Adjacent

y

y = sin(x)

1

3

2

2

1

2 x

y

y = cos(x)

1

3

2

2

1

y

y = tan(x)

1

3

2

2

1

2

x 2 x

LAWS AND IDENTITIES

y

p

1 2

,

3 2

p p

2 2

,

2 2

p

3 2

,

1 2

2

3

3

4

120

5

6

150 135

(0, 1)

2

90

p

1 2

,

3 2

p p

2 2

,

2 2

3

60

4

6

45 30

p

3 2

,

1 2

( 1, 0)

180

(1, 0)

3600 2

x

210 225

315 330

7

11

6

6

p

3 2

,

1 2

5 4

240

270

300

7 4

4

5

p

3 2

,

1 2

p p

3

3

3

p p

2 2

,

2 2

2

2 2

,

2 2

p

1 2

,

3 2

(0, 1)

p

1 2

,

3 2

y

y = csc(x)

1

2

1

3

2

2 x

y

y = sec(x)

1

3

2

2

2 x

1

y

y = cot(x)

1

2

1

3

2

2 x

csusm.edu/stemsc

XXX

csusm_stemcenter

Tel: STEM SC (N): (760) 750-4101 STEM SC (S): (760) 750-7324

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