What is the general solution for sin(15x)+cos(15x)=0 ?

The general solution for sin(15x)+cos(15x)=0 is x= pi/(20)+(pin)/(15)

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Popular Trigonometry >

sin(15x)+cos(15x)=0

Solution

sin (15 x) + cos (15 x) = 0

Solution

x = \frac{π}{20} + \frac{πn}{15}

Degrees

x = 9^{\circ} + 1 2^{\circ} n

Solution steps

sin (15 x) + cos (15 x) = 0

Rewrite using trig identities

sin (15 x) + cos (15 x) = 0

Divide both sides by

cos (15 x), cos (15 x) \neq = 0

\frac{sin ( 15 x ) + cos ( 15 x )}{cos ( 15 x )} = \frac{0}{cos ( 15 x )}

Simplify

\frac{sin ( 15 x )}{cos ( 15 x )} + 1 = 0

Use the basic trigonometric identity:

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

tan (15 x) + 1 = 0

tan (15 x) + 1 = 0

Move

1

to the right side

tan (15 x) + 1 = 0

Subtract

1

from both sides

tan (15 x) + 1 - 1 = 0 - 1

Simplify

tan (15 x) = - 1

tan (15 x) = - 1

General solutions for

tan (15 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}

15 x = \frac{3 π}{4} + πn

15 x = \frac{3 π}{4} + πn

Solve

15 x = \frac{3 π}{4} + πn : x = \frac{π}{20} + \frac{πn}{15}

15 x = \frac{3 π}{4} + πn

Divide both sides by

15

15 x = \frac{3 π}{4} + πn

Divide both sides by

15

\frac{15 x}{15} = \frac{\frac{3 π}{4}}{15} + \frac{πn}{15}

Simplify

\frac{15 x}{15} = \frac{\frac{3 π}{4}}{15} + \frac{πn}{15}

Simplify

\frac{15 x}{15} : x

\frac{15 x}{15}

Divide the numbers:

\frac{15}{15} = 1

= x

Simplify

\frac{\frac{3 π}{4}}{15} + \frac{πn}{15} : \frac{π}{20} + \frac{πn}{15}

\frac{\frac{3 π}{4}}{15} + \frac{πn}{15}

\frac{\frac{3 π}{4}}{15} = \frac{π}{20}

\frac{\frac{3 π}{4}}{15}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 15}

Multiply the numbers:

4 \cdot 15 = 60

= \frac{3 π}{60}

Cancel the common factor:

3

= \frac{π}{20}

= \frac{π}{20} + \frac{πn}{15}

x = \frac{π}{20} + \frac{πn}{15}

x = \frac{π}{20} + \frac{πn}{15}

x = \frac{π}{20} + \frac{πn}{15}

x = \frac{π}{20} + \frac{πn}{15}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(15x)+cos(15x)=0 ?
The general solution for sin(15x)+cos(15x)=0 is x= pi/(20)+(pin)/(15)