Trigonometric Identities
These equations are true for every possible value of the angles A, B, and .
Reciprocal Functions Functions of the sum of two angles
1 1 sin (A + B) = sin A cos B + sin B cos A
sin A = csc A =
csc A sin A cos (A + B) = cos A cos B – sin A sin B
1 1 tan A + tan B
cos A = sec A = tan (A + B) =
sec A cos A 1 – tan A tan B
1 1 Functions of the difference of two angles
tan A = cot A =
cot A 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)
tan A – tan B
tan (A – B) =
1 + tan A tan B
sin A = cos –A cos A = sin –A
2 2
Double Angle Formula
tan A = cot – A cot A = tan –A sin (2A) = 2 sin A cos A
2 2
cos (2A) = cos2 A – sin2 A
sec A = csc –A csc A = sec –A = 1 – 2 sin2 A
2 2
= 2 cos2 A – 1
Negative Angle Relations
2 tan A
sin(– A) = – sin A cos(– A) = cos A tan (2A) =
1 – tan2 A
tan(– A) = – tan A
Power Reduction Formulas
Quotient Relations sin2 A = ½ (1 – cos 2A)
tan2 A = (1 – cos 2A)
sin A cos A cos2 A = ½ (1 + cos 2A) (1 + cos 2A)
tan A = cot A =
cos A sin A
Half Angle Formula
Supplementary Angle Relations The signs of sin
The angles A and B are supplementary 1 – cos A
sin A = ±
A/ and cos A/
2 2
angles such that A + B = . 2 2 depends on the
quadrant in
sin( – A) = sin A cos( – A) = – cos A which A/2 lies.
cos A
1 + cos A
=±
tan( – A) = – tan A 2 2
Pythagorean Identities
1 – cos A
tan A = ±
sin2 A+ cos2 A=1 2 1 + cos A
tan A + 1 = sec2 A
2
1 – cos A sin A
cot2 A + 1 = csc2 A = sin A = 1 + cos A
…\TrigIdentities_FillIn.ppt as of 18 December 2013 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)
A+B A–B
sin A – sin B = 2 cos sin
Sum of two cubes a3 + b3 = (a + b) (a2 – ab + b2) 2 2
A+B A–B
Distribution Property ab + ac = a(b + c) cos A – cos B = –2 sin sin
2 2
Product-to-Sum Formulas A+B A–B
sin A + sin B = 2 sin cos
sin A cos B = ½ [sin (A + B) + sin (A – B)] 2 2
cos A sin B = ½ [sin (A + B) – sin (A – B)] A+B A–B
cos A + cos B = 2 cos cos
cos A cos B = ½ [cos (A + B) + cos (A – B)] 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 sin B sin C
= =
a b c
b2 = a2 + c2 – 2ac cos B
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
