Trigonometric Identities
MVCC Learning Commons IT129
Six Trigonometric Functions
Right triangle definitions, where
0 < 𝜃𝜃<𝜋𝜋/2
sin 𝜃𝜃=opp
hyp
csc 𝜃𝜃=hyp
opp
cos 𝜃𝜃=adj
hyp
sec 𝜃𝜃=hyp
adj
tan 𝜃𝜃=opp
adj
cot 𝜃𝜃=adj
opp
Circular function definitions, where 𝜃𝜃 is any angle.
𝑟𝑟=�𝑥𝑥
2
+𝑦𝑦
2
sin 𝜃𝜃=𝑦𝑦
𝑟𝑟
csc 𝜃𝜃=𝑟𝑟
𝑦𝑦
cos 𝜃𝜃=𝑥𝑥
𝑟𝑟
sec 𝜃𝜃=𝑟𝑟
𝑥𝑥
tan 𝜃𝜃=𝑦𝑦
𝑥𝑥
cot 𝜃𝜃=𝑥𝑥
𝑦𝑦
Negative Angle Identities
sin(−𝜃𝜃)=−sin 𝜃𝜃
cos(−𝜃𝜃)=cos 𝜃𝜃
tan(−𝜃𝜃)=−tan 𝜃𝜃
csc(−𝜃𝜃)=−csc 𝜃𝜃
sec(−𝜃𝜃)=sec 𝜃𝜃
cot(−𝜃𝜃)=−cot 𝜃𝜃
Reciprocal Identities
sin 𝜃𝜃=1
csc 𝜃𝜃
cos 𝜃𝜃=1
sec 𝜃𝜃
tan 𝜃𝜃=1
cot 𝜃𝜃
csc 𝜃𝜃=1
sin 𝜃𝜃
sec 𝜃𝜃=1
cos 𝜃𝜃
cot 𝜃𝜃=1
tan 𝜃𝜃
Tangent and Cotangent Identities
tan 𝜃𝜃=sin 𝜃𝜃
cos 𝜃𝜃
cot 𝜃𝜃=cos 𝜃𝜃
sin 𝜃𝜃
Double Angle Identities
sin 2𝜃𝜃= 2 sin 𝜃𝜃cos 𝜃𝜃
cos 2𝜃𝜃=cos2𝜃𝜃−sin2𝜃𝜃= 2 cos2𝜃𝜃−1 = 1 −2sin2𝜃𝜃
tan 2𝜃𝜃=
2tan 𝜃𝜃
1−tan
2
𝜃𝜃
Pythagorean Identities
sin
2
𝜃𝜃+cos
2
𝜃𝜃= 1 tan
2
𝜃𝜃+ 1 = sec
2
𝜃𝜃
cot2𝜃𝜃+ 1 = csc 2𝜃𝜃
Cofunction Identities
sin �𝜋𝜋
2−𝜃𝜃�=cos 𝜃𝜃
cos �𝜋𝜋
2−𝜃𝜃�=sin 𝜃𝜃
csc �𝜋𝜋
2−𝜃𝜃�=sec 𝜃𝜃
tan �𝜋𝜋
2−𝜃𝜃�=cot 𝜃𝜃
sec �𝜋𝜋
2−𝜃𝜃�=csc 𝜃𝜃
cot �𝜋𝜋
2−𝜃𝜃�=tan 𝜃𝜃
Half Angle Identities
sin 𝜃𝜃
2= ±�1−cos 𝜃𝜃
2 cos 𝜃𝜃
2= ±�1 + cos 𝜃𝜃
2
tan
𝜃𝜃
2= ±�
1−cos 𝜃𝜃
1 + cos 𝜃𝜃
Sum and Difference Formulas
sin 𝐴𝐴+sin 𝐵𝐵= 2 si n �
𝐴𝐴+𝐵𝐵
2�cos �
𝐴𝐴−𝐵𝐵
2�
sin 𝐴𝐴−sin 𝐵𝐵= 2 cos �𝐴𝐴+𝐵𝐵
2�sin �𝐴𝐴−𝐵𝐵
2�
cos 𝐴𝐴+cos 𝐵𝐵= 2 cos �𝐴𝐴+𝐵𝐵
2�cos �𝐴𝐴−𝐵𝐵
2�
cos 𝐴𝐴−cos 𝐵𝐵=−2sin �𝐴𝐴+𝐵𝐵
2�sin �𝐴𝐴−𝐵𝐵
2�
Addition and Subtraction Formulas
sin(𝐴𝐴±𝐵𝐵)=sin 𝐴𝐴cos 𝐵𝐵±cos 𝐴𝐴sin 𝐵𝐵
cos(𝐴𝐴±𝐵𝐵)=cos 𝐴𝐴cos 𝐵𝐵∓sin 𝐴𝐴sin 𝐵𝐵
tan(𝐴𝐴±𝐵𝐵)=
tan 𝐴𝐴±tan 𝐵𝐵
1∓tan 𝐴𝐴tan 𝐵𝐵
Product Formulas
sin 𝐴𝐴sin 𝐵𝐵=
1
2[cos(𝐴𝐴−𝐵𝐵)−cos(𝐴𝐴+𝐵𝐵)]
cos 𝐴𝐴cos 𝐵𝐵=1
2[cos(𝐴𝐴+𝐵𝐵)+cos(𝐴𝐴−𝐵𝐵)]
sin 𝐴𝐴cos 𝐵𝐵=1
2[sin(𝐴𝐴+𝐵𝐵)+sin(𝐴𝐴−𝐵𝐵)]
cos 𝐴𝐴sin 𝐵𝐵=1
2[sin(𝐴𝐴+𝐵𝐵)−sin(𝐴𝐴−𝐵𝐵)]
