Write the identies of trigonometry?
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Answers
Sine Function: sin(θ) = Opposite / Hypotenuse
Cosine Function: cos(θ) = Adjacent / Hypotenuse
Tangent Function: tan(θ) = Opposite / Adjacent
Recall that these identities work both ways. That is, if you have an expression that matches the left or right side of an identity, you can replace it with whatever is on the other side.
A. Reciprocal identities
a1 sin A =
1
csc A
a4 csc A =
1
sin A
a2 cos A =
1
sec A
a5 sec A =
1
cos A
a3 tan A =
1
cot A
a6 cot A =
1
tan A
B. Ratio identities
b1 tan A =
sin A
cos A
b2 cot A =
cos A
sin A
C. Opposite Angle identities
c1 sin ( − A ) = − sin A
c2 cos ( − A ) = + c o s A
c3 tan ( − A ) = − tan A
D. Pythagorean identities
d1 sin 2 A + cos 2 A = 1
d2 1 + tan 2 A = sec 2 A
d3 1 + cot 2 A = csc 2 A
E. Complementary angle identities
e1 sin A = cos
π
2
− A
e2 tan A = cot
π
2
− A
e3 sec A = csc
π
2
− A
* Note:
π
2
is 90° in radians.
If A is in degrees, use 90 instead of
π
2
For example: sin A = cos ( 90 − A )
F. Supplementary angle identities
This basically says that if two angles are supplementary (add to 180°) they have the same sine.
f1 sin A = s i n ( π − A ) ( 0 < = A < = π )
Or in degrees:
sin A = s i n ( 180 − A ) ( 0 < = A < = 180 )
G. The sum identities
g1 sin ( A + B ) = sin A cos B + cos A sin B
g2 cos ( A + B ) = cos A cos B − sin A sin B
g3 tan ( A + B ) =
tan A + tan B
1 − tan A tan B
H. The difference identities
h1 sin ( A − B ) = sin A cos B − cos A sin B
h2 cos ( A − B ) = cos A cos B + sin A sin B
h3 tan ( A − B ) =
tan A − tan B
1 − tan A tan B
J. The double angle identities
j1 sin 2 A = 2 sin A cos A
j2 cos 2 A = cos 2 A − sin 2 A
j3 cos 2 A = 2 cos 2 A − 1
j4 cos 2 A = 1 − 2 sin 2 A
j5 tan 2 A =
2 tan A
1 − tan 2 A
K. The half angle identities
k1 sin
A
2
= √
1 − cos A
2
k2 cos
A
2
= √
1 + cos A
2
k3 tan
A
2
=
sin A
1 + cos A
k4 tan
A
2
=
1 − cos A
sin A
M. The sine identities
These show how to represent the sine function in terms of the other five functions. Some of these identities may also appear under other headings.
m1 sin A = ± √ 1 − cos 2 A
m2 sin A = ±
tan A
√ 1 + tan 2 A
m3 sin A =
1
csc A
m4 sin A = ±
√ sec 2 A − 1
sec A
m5 sin A = ±
1
√ 1 + cot 2 A
N. The cosine identities
These show how to represent the cosine function in terms of the other five functions. Some of these identities may also appear under other headings.
n1 cos A = ± √ 1 − sin 2 A
n2 cos A = ±
1
√ 1 + tan 2 A
n3 cos A =
1
sec A
n4 cos A = ±
√ csc 2 A − 1
csc A
n5 cos A = ±
cot A
√ 1 + cot 2 A
P. The tangent identities
These show how to represent the tangent function in terms of the other five functions. Some of these trig identities may also appear under other headings.
p1 tan A = ± √ sec 2 − 1
p2 tan A = ±
1
√ csc 2 A − 1
p3 tan A =
1
cot A
p4 tan A = ±
√ 1 − cos 2 A
cos A
p5 tan A = ±
sin A
√ 1 − sin 2 A