Trigonometry Identities

Basic trig identities you may find useful in Calc 1

\frac{1}{csc (u)} =

sin (u)

\frac{1}{sec (u)} =

cos (u)

\frac{1}{cot (u)} =

tan (u)

\frac{1}{sin (u)} =

csc (u)

\frac{1}{cos (u)} =

sec (u)

\frac{1}{tan (u)} =

cot (u)

\frac{sin (u)}{cos (u)} =

tan (u)

\frac{cos (u)}{sin (u)} =

cot (u)

sin²(u) + cos²(u) =

1

1 + tan²(u) =

sec²(x)

1 + cot²(u) =

csc²(u)

1 - cos²(u) =

sin²(u)

1 - sin²(u) =

cos²(u)

sec²(u) - 1 =

tan²(u)

sin(\frac{π}{2} - u) =

cos(u)

cos(\frac{\pi}{2} - u) =

sin(u)

tan(\frac{\pi}{2} - u) =

cot(u)

sin(-u) =

-sin(u)

csc(-u) =

-csc(u)

cos(-u) =

cos(u)

sec(-u) =

sec(u)

tan(-u) =

-tan(u)

cot(-u) =

-cot(u)

sin(2u) =

2*sin(u)*cos(u)

cos(2u) =

cos²(u)-sin²(u)

2cos²(u)-1

1-2sin²(u)

tan(2u) =

\frac{2tan(u)}{1-tan²(u)}

sin²(u) =

\frac{1-cos(2u)}{2}

sin(\frac{u}{2}) =

\pm\sqrt{\frac{1-cos(u)}{2}}

cos²(u) =

\frac{1+cos(2u)}{2}

\cos(\frac{u}{2})=

\pm\sqrt{\frac{1+cos(u)}{2}}

tan(\frac{u}{2})

\pm\sqrt{\frac{1-cos(u)}{1+cos(u)}}

\tan^2\left(u\right)=

\frac{1-\cos\left(2u\right)}{1+\cos\left(2u\right)}

sin\left(u\right)=\frac{?}{?}

sin\left(u\right)=\frac{opp.}{hyp.}=\frac{y}{r}

\cos\left(u\right)=\frac{?}{?}

\cos\left(u\right)=\frac{adj.}{hyp.}=\frac{x}{r}

\tan\left(u\right)=\frac{?}{?}

\tan\left(u\right)=\frac{opp.}{adj.}=\frac{y}{x}

\csc\left(u\right)=\frac{?}{?}

\csc\left(u\right)=\frac{hyp.}{opp.}=\frac{r}{y}

\sec\left(u\right)=\frac{?}{?}

\sec\left(u\right)=\frac{hyp.}{adj.}=\frac{r}{x}

\cot\left(u\right)=\frac{?}{?}

\cot\left(u\right)=\frac{adj.}{opp.}=\frac{x}{y}

\left\vert r\right\vert=?

\left\vert r\right\vert=\sqrt{x^2+y^2}