What is the general solution for sin(pix)+cos(pix)=0 ?

The general solution for sin(pix)+cos(pix)=0 is x= 3/4+n

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sin(pix)+cos(pix)=0

Solution

sin (π x) + cos (π x) = 0

Solution

x = \frac{3}{4} + n

Degrees

x = 42.97183 \dots^{\circ}

Solution steps

sin (π x) + cos (π x) = 0

Rewrite using trig identities

sin (π x) + cos (π x) = 0

Divide both sides by

cos (π x), cos (π x) \neq = 0

\frac{sin ( π x ) + cos ( π x )}{cos ( π x )} = \frac{0}{cos ( π x )}

Simplify

\frac{sin ( π x )}{cos ( π x )} + 1 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (π x) + 1 = 0

tan (π x) + 1 = 0

Move

1

to the right side

tan (π x) + 1 = 0

Subtract

1

from both sides

tan (π x) + 1 - 1 = 0 - 1

Simplify

tan (π x) = - 1

tan (π x) = - 1

General solutions for

tan (π x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

π x = \frac{3 π}{4} + πn

π x = \frac{3 π}{4} + πn

Solve

π x = \frac{3 π}{4} + πn : x = \frac{3}{4} + n

π x = \frac{3 π}{4} + πn

Divide both sides by

π

π x = \frac{3 π}{4} + πn

Divide both sides by

π

\frac{π x}{π} = \frac{\frac{3 π}{4}}{π} + \frac{πn}{π}

Simplify

\frac{π x}{π} = \frac{\frac{3 π}{4}}{π} + \frac{πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{3 π}{4}}{π} + \frac{πn}{π} : \frac{3}{4} + n

\frac{\frac{3 π}{4}}{π} + \frac{πn}{π}

\frac{\frac{3 π}{4}}{π} = \frac{3}{4}

\frac{\frac{3 π}{4}}{π}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3 π}{4 π}

Cancel the common factor:

π

= \frac{3}{4}

\frac{πn}{π} = n

\frac{πn}{π}

Cancel the common factor:

π

= n

= \frac{3}{4} + n

x = \frac{3}{4} + n

x = \frac{3}{4} + n

x = \frac{3}{4} + n

x = \frac{3}{4} + n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(pix)+cos(pix)=0 ?
The general solution for sin(pix)+cos(pix)=0 is x= 3/4+n