Basic Trigonometric Identities - Bucks County Community College

Basic Trigonometric Identities

Reciprocals

sin() = 1

csc()

cos()

=

1 sec()

tan()

=

1 cot()

csc() = 1

sin()

sec()

=

1 cos()

cot() =

1 tan()

Pythagorean sin2(x) + cos2(x) = 1

1 + tan2(x) = sec2(x)

cot2(x) + 1 = csc2(x)

Negative Angles sin(-) = -sin()

Quotient Identities tan() = sin()

cos()

cos(-) = cos()

tan(-) = -tan()

cot() = cos()

sin()

Sum and Difference Formulas sin( + ) = sin() cos() + sin() cos()

cos( + ) = cos(t) cos(u) - sin(t) sin(u)

tan( + ) =

tan()+tan() 1-tan()tan()

sin( - ) = sin() cos() - sin() cos()

cos( - ) = cos(t) cos(u) + sin(t) sin(u)

tan(

-

)

=

tan()-tan() 1+tan()tan()

Double Angles sin(2t) = 2 sin(t) cos(t)

cos(2t) = cos2(t) - sin2(t) = 2cos2(t) - 1 = 1 - 2sin2(t)

tan(2)

=

2tan () 1-2()

BCCC ASC Rev. 6/2019

Cofunction Identities cos(90? - ) = sin() cot(90? - ) = tan()

-over-

sin(90? - ) = cos() sec(90? - ) = csc()

tan(90? - ) = cot() csc(90? - ) = sec()

Half Angles

sin () = ?1-cos()

2

2

cos () = ?1+cos()

2

2

tan

()

2

=

?11-+ccooss(())

= 1-cos()

sin()

= ()

1+()

Product-to-Sum sin() cos() = 1 [sin( + ) + sin( - )]

2

cos() cos() = 1 [cos( + ) + cos( - )]

2

cos() sin() = 1 [sin( + ) - sin( - )]

2

sin() sin() = 1 [cos( - ) - cos( + )]

2

Sum-to-Product

sin(t) + sin(u) = 2 sin (t+u) cos (t-u)

2

2

sin(t) - sin(u) = 2 cos (t+u) sin (t-u)

2

2

cos(t) + cos(u) = 2 cos (t+u) cos (t-u)

2

2

cos(t) + cos(u) = -2 sin (t+u) sin (t-u)

2

2

Law of Sines = =

sin() sin() sin()

Laws of Cosines 2 = 2 + 2 - 2 cos() 2 = 2 + 2 - 2 cos() 2 = 2 + 2 - 2 cos()

B

a

c

C

A

b

BCCC ASC Rev. 6/2019

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