NMT / ZNO

Formulas and reference

A compact mathematics reference in two levels: the NMT exam sheet and the full ZNO formula set.

ZNO — full reference · Algebra · 19 formulas
Full ZNO Expanded school formula set. The “NMT” badge marks formulas also present on the compact sheet.
Trigonometry

Trigonometry

Fundamental trigonometric identity

NMT
\[\sin^2\alpha+\cos^2\alpha=1\]
10–11 grade

Tangent

NMT
\[\tg\alpha=\frac{\sin\alpha}{\cos\alpha}\]
Condition \(\cos\alpha\ne0\)
10–11 grade

Cotangent

\[\ctg\alpha=\frac{\cos\alpha}{\sin\alpha}\]
Condition \(\sin\alpha\ne0\)
10–11 grade

Tangent identity

NMT
\[1+\tg^2\alpha=\frac1{\cos^2\alpha}\]
Condition \(\cos\alpha\ne0\)
10–11 grade

Cotangent identity

\[1+\ctg^2\alpha=\frac1{\sin^2\alpha}\]
Condition \(\sin\alpha\ne0\)
10–11 grade

Sine of a sum

\[\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\]
10–11 grade

Sine of a difference

\[\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\]
10–11 grade

Cosine of a sum

\[\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\]
10–11 grade

Cosine of a difference

\[\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\]
10–11 grade

Sine of a double angle

NMT
\[\sin2\alpha=2\sin\alpha\cos\alpha\]
10–11 grade

Cosine of a double angle

NMT
\[\cos2\alpha=\cos^2\alpha-\sin^2\alpha\]
\[\cos2\alpha=2\cos^2\alpha-1\]
\[\cos2\alpha=1-2\sin^2\alpha\]
10–11 grade

Tangent of a double angle

\[\tg2\alpha=\frac{2\tg\alpha}{1-\tg^2\alpha}\]
10–11 grade

Square of sine of a half-angle

\[\sin^2\frac\alpha2=\frac{1-\cos\alpha}{2}\]
10–11 grade

Square of cosine of a half-angle

\[\cos^2\frac\alpha2=\frac{1+\cos\alpha}{2}\]
10–11 grade

Formulas for 90°+α

NMT
\[\sin(90^\circ+\alpha)=\cos\alpha\]
\[\cos(90^\circ+\alpha)=-\sin\alpha\]
\[\tg(90^\circ+\alpha)=-\frac1{\tg\alpha}\]
10–11 grade

Formulas for 180°−α

NMT
\[\sin(180^\circ-\alpha)=\sin\alpha\]
\[\cos(180^\circ-\alpha)=-\cos\alpha\]
\[\tg(180^\circ-\alpha)=-\tg\alpha\]
10–11 grade
Trigonometry

Trigonometric equations

Equation sin x=a

\[x=(-1)^n\arcsin a+\pi n,\quad n\in\mathbb Z\]
Condition \(|a|\le1\)
10–11 grade

Equation cos x=a

\[x=\pm\arccos a+2\pi n,\quad n\in\mathbb Z\]
Condition \(|a|\le1\)
10–11 grade

Equation tg x=a

\[x=\arctg a+\pi n,\quad n\in\mathbb Z\]
10–11 grade