Tuần hoàn, đối xứng và tịnh tiến Đẳng_thức_lượng_giác

Các đẳng thức sau có thể dễ thấy trên vòng tròn đơn vị:

Tuần hoàn (k nguyên)Đối nhau:Phụ nhauBù nhauHơn kém nhau π {\displaystyle \pi } Hơn kém nhau π 2 {\displaystyle {\frac {\pi }{2}}}
sin ⁡ ( x ) = sin ⁡ ( x + 2 k π ) {\displaystyle \sin(x)=\sin(x+2k\pi )\,} sin ⁡ ( − x ) = − sin ⁡ ( x ) {\displaystyle \sin(-x)=-\sin(x)\,} sin ⁡ ( x ) = cos ⁡ ( π 2 − x ) {\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)} sin ⁡ ( π − x ) = sin ⁡ ( x ) {\displaystyle \sin(\pi -x)=\sin(x)} sin ⁡ ( π + x ) = − sin ⁡ ( x ) {\displaystyle \sin(\pi +x)=-\sin(x)} sin ⁡ ( x ) = − cos ⁡ ( π 2 + x ) {\displaystyle \sin(x)=-\cos \left({\frac {\pi }{2}}+x\right)}
cos ⁡ ( x ) = cos ⁡ ( x + 2 k π ) {\displaystyle \cos(x)=\cos(x+2k\pi )\,} cos ⁡ ( − x ) = cos ⁡ ( x ) {\displaystyle \cos(-x)=\;\cos(x)\,} cos ⁡ ( x ) = sin ⁡ ( π 2 − x ) {\displaystyle \cos(x)=\sin \left({\frac {\pi }{2}}-x\right)} cos ⁡ ( π − x ) = − cos ⁡ ( x ) {\displaystyle \cos(\pi -x)=\;-\cos(x)\,} cos ⁡ ( π + x ) = − cos ⁡ ( x ) {\displaystyle \cos(\pi +x)=\;-\cos(x)\,} cos ⁡ ( x ) = sin ⁡ ( π 2 + x ) {\displaystyle \cos(x)=\sin \left({\frac {\pi }{2}}+x\right)}
tan ⁡ ( x ) = tan ⁡ ( x + k π ) {\displaystyle \tan(x)=\tan(x+k\pi )\,} tan ⁡ ( − x ) = − tan ⁡ ( x ) {\displaystyle \tan(-x)=-\tan(x)\,} tan ⁡ ( x ) = cot ⁡ ( π 2 − x ) {\displaystyle \tan(x)=\cot \left({\frac {\pi }{2}}-x\right)} tan ⁡ ( π − x ) = − tan ⁡ ( x ) {\displaystyle \tan(\pi -x)=-\tan(x)\,} tan ⁡ ( π + x ) = tan ⁡ ( x ) {\displaystyle \tan(\pi +x)=\tan(x)\,} tan ⁡ ( x ) = − cot ⁡ ( π 2 + x ) {\displaystyle \tan(x)=-\cot \left({\frac {\pi }{2}}+x\right)}
cot ⁡ ( x ) = cot ⁡ ( x + k π ) {\displaystyle \cot(x)=\cot(x+k\pi )} cot ⁡ ( − x ) = − cot ⁡ ( x ) {\displaystyle \cot(-x)=-\cot(x)\,} cot ⁡ ( x ) = tan ⁡ ( π 2 − x ) {\displaystyle \cot(x)=\tan \left({\frac {\pi }{2}}-x\right)} cot ⁡ ( π − x ) = − cot ⁡ ( x ) {\displaystyle {\displaystyle \cot(\pi -x)=-\cot(x)\,}} cot ⁡ ( π + x ) = cot ⁡ ( x ) {\displaystyle {\displaystyle \cot(\pi +x)=\cot(x)\,}} cot ⁡ ( x ) = − tan ⁡ ( π 2 + x ) {\displaystyle \cot(x)=-\tan \left({\frac {\pi }{2}}+x\right)}

Đẳng thức sau cũng đôi khi hữu ích: a sin ⁡ x + b cos ⁡ x = a 2 + b 2 ⋅ sin ⁡ ( x + φ ) {\displaystyle a\sin x+b\cos x={\sqrt {a^{2}+b^{2}}}\cdot \sin(x+\varphi )}

với φ = { arctan ⁡ b a , n e ^ ´ u   a ≥ 0 ; π + arctan ⁡ b a , n e ^ ´ u   a < 0. {\displaystyle \varphi =\left\{{\begin{matrix}\arctan {\dfrac {b}{a}},&&{\mbox{n}}{\acute {\hat {\mbox{e}}}}{\mbox{u}}\ a\geq 0;\\\pi +\arctan {\dfrac {b}{a}},&&{\mbox{n}}{\acute {\hat {\mbox{e}}}}{\mbox{u}}\ a<0.\end{matrix}}\right.}