What is the general solution for sin(θ)=-cos(θ) ?

The general solution for sin(θ)=-cos(θ) is θ=(3pi)/4+pin

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sin(θ)=-cos(θ)

sin (θ) = - cos (θ)

θ = \frac{3 π}{4} + πn

Degrees

θ = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ) = - cos (θ)

Rewrite using trig identities

sin (θ) = - cos (θ)

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{sin ( θ )}{cos ( θ )} = \frac{- cos ( θ )}{cos ( θ )}

Simplify

\frac{sin ( θ )}{cos ( θ )} = - 1

Use the basic trigonometric identity:

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

tan (θ) = - 1

tan (θ) = - 1

General solutions for

tan (θ) = - 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}

θ = \frac{3 π}{4} + πn

θ = \frac{3 π}{4} + πn

cot^{2} (θ) + csc (θ) = 1

5 cos^{2} (θ) - 2 cos (θ) = 0

tan^{2} (x) - 2 tan (x) - 3 = 0

6 cos (x) - 3 = 0

cos^{2} (x) - 1 = 0, 0 \leq x \leq 2 π

What is the general solution for sin(θ)=-cos(θ) ?
The general solution for sin(θ)=-cos(θ) is θ=(3pi)/4+pin