What is the value of (sin(pi)+picos(pi))/(cos(pi)-pisin(pi)) ?

The value of (sin(pi)+picos(pi))/(cos(pi)-pisin(pi)) is pi

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(sin(pi)+picos(pi))/(cos(pi)-pisin(pi))

\frac{sin ( π ) + π cos ( π )}{cos ( π ) - π sin ( π )}

π

Decimal Notation

3.14159 \dots

Solution steps

\frac{sin ( π ) + π cos ( π )}{cos ( π ) - π sin ( π )}

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

= \frac{0 + π ( - 1 )}{- 1 - π 0}

Simplify

= π

cos (21 5^{\circ})

cos (π^{3})

cos (π^{0})

arctan (3) - arctan (1)

sin (2 arccos (- \frac{3}{2}))

What is the value of (sin(pi)+picos(pi))/(cos(pi)-pisin(pi)) ?
The value of (sin(pi)+picos(pi))/(cos(pi)-pisin(pi)) is pi