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sin(2x)=-5cos(2x),0<= x<= 2pi

Solution

sin (2 x) = - 5 cos (2 x), 0 \leq x \leq 2 π

x = \frac{- 1.37340 \dots + π}{2}, x = \frac{- 1.37340 \dots + 2 π}{2}, x = \frac{- 1.37340 \dots + 3 π}{2}, x = \frac{- 1.37340 \dots + 4 π}{2}

Degrees

x = 50.65496 \dots^{\circ}, x = 140.65496 \dots^{\circ}, x = 230.65496 \dots^{\circ}, x = 320.65496 \dots^{\circ}

Solution steps

sin (2 x) = - 5 cos (2 x), 0 \leq x \leq 2 π

Rewrite using trig identities

sin (2 x) = - 5 cos (2 x)

Divide both sides by

cos (2 x), cos (2 x) \neq = 0

\frac{sin ( 2 x )}{cos ( 2 x )} = \frac{- 5 cos ( 2 x )}{cos ( 2 x )}

Simplify

\frac{sin ( 2 x )}{cos ( 2 x )} = - 5

Use the basic trigonometric identity:

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

tan (2 x) = - 5

tan (2 x) = - 5

Apply trig inverse properties

tan (2 x) = - 5

General solutions for

tan (2 x) = - 5

tan (x) = - a \Rightarrow x = arctan (- a) + πn

2 x = arctan (- 5) + πn

2 x = arctan (- 5) + πn

Solve

2 x = arctan (- 5) + πn : x = - \frac{arctan ( 5 )}{2} + \frac{πn}{2}

2 x = arctan (- 5) + πn

Simplify

arctan (- 5) + πn : - arctan (5) + πn

arctan (- 5) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 5) = - arctan (5)

= - arctan (5) + πn

2 x = - arctan (5) + πn

Divide both sides by

2

2 x = - arctan (5) + πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arctan ( 5 )}{2} + \frac{πn}{2}

Simplify

x = - \frac{arctan ( 5 )}{2} + \frac{πn}{2}

x = - \frac{arctan ( 5 )}{2} + \frac{πn}{2}

x = - \frac{arctan ( 5 )}{2} + \frac{πn}{2}

Solutions for the range

0 \leq x \leq 2 π

x = \frac{- arctan ( 5 ) + π}{2}, x = \frac{- arctan ( 5 ) + 2 π}{2}, x = \frac{- arctan ( 5 ) + 3 π}{2}, x = \frac{- arctan ( 5 ) + 4 π}{2}

Show solutions in decimal form

x = \frac{- 1.37340 \dots + π}{2}, x = \frac{- 1.37340 \dots + 2 π}{2}, x = \frac{- 1.37340 \dots + 3 π}{2}, x = \frac{- 1.37340 \dots + 4 π}{2}

2 cos^{2} (x) + 9 cos (x) - 5 = 0

cos (θ) = - \frac{3}{4}

tan (θ) = \frac{3}{2}

5 sin (θ) + 7 = 0

5 sin (x) tan (x) + 4 tan (x) = 0