What is the general solution for sin(2θ)=-24/25 ?

The general solution for sin(2θ)=-24/25 is θ=-(1.28700…)/2+pin,θ= pi/2+(1.28700…)/2+pin

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sin(2θ)=-24/25

sin (2 θ) = - \frac{24}{25}

θ = - \frac{1.28700 \dots}{2} + πn, θ = \frac{π}{2} + \frac{1.28700 \dots}{2} + πn

Degrees

θ = - 36.86989 \dots^{\circ} + 18 0^{\circ} n, θ = 126.86989 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 θ) = - \frac{24}{25}

Apply trig inverse properties

sin (2 θ) = - \frac{24}{25}

General solutions for

sin (2 θ) = - \frac{24}{25}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

2 θ = arcsin (- \frac{24}{25}) + 2 πn, 2 θ = π + arcsin (\frac{24}{25}) + 2 πn

2 θ = arcsin (- \frac{24}{25}) + 2 πn, 2 θ = π + arcsin (\frac{24}{25}) + 2 πn

Solve

2 θ = arcsin (- \frac{24}{25}) + 2 πn : θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn

2 θ = arcsin (- \frac{24}{25}) + 2 πn

Simplify

arcsin (- \frac{24}{25}) + 2 πn : - arcsin (\frac{24}{25}) + 2 πn

arcsin (- \frac{24}{25}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{24}{25}) = - arcsin (\frac{24}{25})

= - arcsin (\frac{24}{25}) + 2 πn

2 θ = - arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

2 θ = - arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = - \frac{arcsin ( \frac{24}{25} )}{2} + \frac{2 πn}{2}

Simplify

θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn

θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn

Solve

2 θ = π + arcsin (\frac{24}{25}) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

2 θ = π + arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

2 θ = π + arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

Show solutions in decimal form

θ = - \frac{1.28700 \dots}{2} + πn, θ = \frac{π}{2} + \frac{1.28700 \dots}{2} + πn

sin (4 x) + cos (4 x) = 1

- 4 cos (x) = - sin^{2} (x) + 4, 0 \leq x \leq 2 π

0 = cos^{2} (\frac{π}{12} t)

sec^{2} (x) + 2 sec (x) + 4 = 0

cos (A) = 0.7865

What is the general solution for sin(2θ)=-24/25 ?
The general solution for sin(2θ)=-24/25 is θ=-(1.28700…)/2+pin,θ= pi/2+(1.28700…)/2+pin