Download Trigonometric Identities

Trigonometric Identities
These equations are true for every possible value of the angles A, B, and .
Functions of the sum of two angles
Reciprocal Functions
sin A =
1
csc A
csc A =
1
sin A
sin (A + B) = sin A cos B + sin B cos A
cos (A + B) = cos A cos B – sin A sin B
tan A + tan B
1 – tan A tan B
cos A =
1
sec A
sec A =
1
cos A
tan (A + B) =
1
cot A
cot A =
1
tan A
Functions of the difference of two angles
tan A =
sin (A – B) = sin A cos B – sin B cos A
cos (A – B) = cos A cos B + sin A sin B
Co-functions (radian form)
 
sin A = cos 
2

–A
 
cos A = sin 
2
 
tan A = cot 
2

– A
 

cot A = tan 
–A
2
 
sec A = csc
2

–A
 
csc A = sec
2

–A
tan A – tan B
1 + tan A tan B
tan (A – B) =
Double Angle Formula
sin (2A) = 2 sin A cos A
cos (2A) = cos2 A – sin2 A

–A
= 1 – 2 sin2 A
= 2 cos2 A – 1
Negative Angle Relations
sin(– A) = – sin A
tan(– A) = – tan A
cos(– A) = cos A
Quotient Relations
tan A =
sin A
cos A
cot A =
cos A
sin A
Supplementary Angle Relations
The angles A and B are supplementary
angles such that A + B = .
sin( – A) = sin A
tan( – A) = – tan A
cos( – A) = – cos A
Pythagorean Identities
sin2
cos2
A+
A=1
2
tan A + 1 = sec2 A
cot2 A + 1 = csc2 A
…\TrigIdentities_FillIn.ppt as of 18 December 2013
2 tan A
1 – tan2 A
tan (2A) =
Power Reduction Formulas
sin2 A = ½ (1 – cos 2A)
tan2 A = (1 – cos 2A)
cos2 A = ½ (1 + cos 2A)
(1 + cos 2A)
Half Angle Formula
sin  A  = ±
2
cos  A 
2
=±
tan  A  = ±
2
=

1 – cos A
2

1 + cos A
2

1 – cos A
1 + cos A
1 – cos A
sin A
=
The signs of sin
A/ and cos A/
2
2
depends on the
quadrant in
which A/2 lies.
sin A
1 + cos A
From: Trigonometry: The Easy Way, ISBN 0-8120-4389-8
Trigonometric Identities
Difference of two squares a2 – b2 = (a – b) (a + b)
Sum-to-Product Formulas
Difference of two cubes
a3
–
b3
= (a – b)
(a2
+ ab +
b2)
sin A – sin B =
2 cos
Sum of two cubes a3 + b3 = (a + b) (a2 – ab + b2)
Product-to-Sum Formulas
A+B
A–B
cos
2
2
sin A + sin B = 2 sin
sin A cos B =
½ [sin (A + B) + sin (A – B)]
cos A sin B =
½ [sin (A + B) – sin (A – B)]
cos A cos B =
½ [cos (A + B) + cos (A – B)]
A–B
2
A+B
A–B
sin
2
2
cos A – cos B = –2 sin
Distribution Property ab + ac = a(b + c)
A+B
sin
2
cos A + cos B = 2 cos
A+B
A–B
cos
2
2
sin A sin B = – ½ [cos (A + B) – cos (A – B)]
It is important to note that these identities are only true provided that all the arguments for
the trigonometric functions have permissible values. For example, any identity involving
the tangent function will be unusable if one of the angles has the value of 90 o or /2 .
Reference the Law of Cosines Triangle for the Law of Sines and Cosines formulas.
Law of Cosines
Law of Sines
a2 = b2 + c2 – 2bc cos A
sin A
=
a
b2 = a2 + c2 – 2ac cos B
sin B
b
=
sin C
c
c2 = a2 + b2 – 2ab cos C
B
c
a
A
C
b
Trigonometric Triangular Area
Heron’s Formula
Area = 1/2 ab sin C
Area =  s (s – a) (s – b) (s – c)
Area = 1/2 ac sin B
where s = 1/2 (a + b + c)
Area = 1/2 bc sin A
The sides, a, b, and c, are the sides that make
the angle. Angle C is formed by sides a and b.
…\TrigIdentities_FillIn.ppt as of 18 December 2013
From: Trigonometry: The Easy Way, ISBN 0-8120-4389-8