4=5+3sin(2t-pi/5)

Solution

4 = 5 + 3 sin (2 t - \frac{π}{5})

t = πn + \frac{π}{10} - \frac{0.33983 \dots}{2}, t = πn + \frac{π}{2} + \frac{π}{10} + \frac{0.33983 \dots}{2}

Solution steps

4 = 5 + 3 sin (2 t - \frac{π}{5})

Switch sides

5 + 3 sin (2 t - \frac{π}{5}) = 4

Move

5

to the right side

3 sin (2 t - \frac{π}{5}) = - 1

Divide both sides by

3

sin (2 t - \frac{π}{5}) = - \frac{1}{3}

Apply trig inverse properties

2 t - \frac{π}{5} = arcsin (- \frac{1}{3}) + 2 πn, 2 t - \frac{π}{5} = π + arcsin (\frac{1}{3}) + 2 πn

Solve

2 t - \frac{π}{5} = arcsin (- \frac{1}{3}) + 2 πn : t = πn + \frac{π}{10} - \frac{arcsin ( \frac{1}{3} )}{2}

Solve

2 t - \frac{π}{5} = π + arcsin (\frac{1}{3}) + 2 πn : t = πn + \frac{π}{2} + \frac{π}{10} + \frac{arcsin ( \frac{1}{3} )}{2}

t = πn + \frac{π}{10} - \frac{arcsin ( \frac{1}{3} )}{2}, t = πn + \frac{π}{2} + \frac{π}{10} + \frac{arcsin ( \frac{1}{3} )}{2}

Show solutions in decimal form

t = πn + \frac{π}{10} - \frac{0.33983 \dots}{2}, t = πn + \frac{π}{2} + \frac{π}{10} + \frac{0.33983 \dots}{2}

sin^{2} (2 x) + 2 \cdot cos^{2} (x) - 2 = 0

tan (θ) = - 0.5586, 0 \leq θ < 2 π

6 cos^{2} (x) - sin (x) - 5 = 0, 0 \leq x \leq 2 π

sin (4 x) = sin (3 x)

4 cos (x) + 3 = 5