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Popular >

solvefor θ,r=vcos(θ)((2vsin(θ))/g)

Solution

solve for

θ, r = v cos (θ) (\frac{2 v sin ( θ )}{g})

Solution

θ = \frac{arcsin ( \frac{g r}{v ^{2}} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( - \frac{g r}{v ^{2}} )}{2} + πn

Solution steps

r = v cos (θ) (\frac{2 v sin ( θ )}{g})

Switch sides

v cos (θ) \frac{2 v sin ( θ )}{g} = r

Subtract

r

from both sides

\frac{2 v ^{2} sin ( θ ) cos ( θ )}{g} - r = 0

Simplify

\frac{2 v ^{2} sin ( θ ) cos ( θ )}{g} - r : \frac{2 v ^{2} sin ( θ ) cos ( θ ) - g r}{g}

\frac{2 v ^{2} sin ( θ ) cos ( θ )}{g} - r

Convert element to fraction:

r = \frac{r g}{g}

= \frac{2 v ^{2} sin ( θ ) cos ( θ )}{g} - \frac{r g}{g}

Since the denominators are equal, combine the fractions:

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

= \frac{2 v ^{2} sin ( θ ) cos ( θ ) - r g}{g}

\frac{2 v ^{2} sin ( θ ) cos ( θ ) - g r}{g} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 v^{2} sin (θ) cos (θ) - g r = 0

Rewrite using trig identities

2 v^{2} sin (θ) cos (θ) - g r

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= - g r + sin (2 θ) v^{2}

- g r + sin (2 θ) v^{2} = 0

Move

g r

to the right side

- g r + sin (2 θ) v^{2} = 0

Add

g r

to both sides

- g r + sin (2 θ) v^{2} + g r = 0 + g r

Simplify

sin (2 θ) v^{2} = g r

sin (2 θ) v^{2} = g r

Divide both sides by

v^{2}; v \neq = 0

sin (2 θ) v^{2} = g r

Divide both sides by

v^{2}; v \neq = 0

\frac{sin ( 2 θ ) v ^{2}}{v ^{2}} = \frac{g r}{v ^{2}}; v \neq = 0

Simplify

sin (2 θ) = \frac{g r}{v ^{2}}; v \neq = 0

sin (2 θ) = \frac{g r}{v ^{2}}; v \neq = 0

Apply trig inverse properties

sin (2 θ) = \frac{g r}{v ^{2}}

General solutions for

sin (2 θ) = \frac{g r}{v ^{2}}

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

2 θ = arcsin (\frac{g r}{v ^{2}}) + 2 πn, 2 θ = π + arcsin (- \frac{g r}{v ^{2}}) + 2 πn

2 θ = arcsin (\frac{g r}{v ^{2}}) + 2 πn, 2 θ = π + arcsin (- \frac{g r}{v ^{2}}) + 2 πn

Solve

2 θ = arcsin (\frac{g r}{v ^{2}}) + 2 πn : θ = \frac{arcsin ( \frac{g r}{v ^{2}} )}{2} + πn

2 θ = arcsin (\frac{g r}{v ^{2}}) + 2 πn

Divide both sides by

2

2 θ = arcsin (\frac{g r}{v ^{2}}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{arcsin ( \frac{g r}{v ^{2}} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{arcsin ( \frac{g r}{v ^{2}} )}{2} + πn

θ = \frac{arcsin ( \frac{g r}{v ^{2}} )}{2} + πn

Solve

2 θ = π + arcsin (- \frac{g r}{v ^{2}}) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( - \frac{g r}{v ^{2}} )}{2} + πn

2 θ = π + arcsin (- \frac{g r}{v ^{2}}) + 2 πn

Divide both sides by

2

2 θ = π + arcsin (- \frac{g r}{v ^{2}}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( - \frac{g r}{v ^{2}} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} + \frac{arcsin ( - \frac{g r}{v ^{2}} )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( - \frac{g r}{v ^{2}} )}{2} + πn

θ = \frac{arcsin ( \frac{g r}{v ^{2}} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( - \frac{g r}{v ^{2}} )}{2} + πn

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