What is the general solution for 73500=130000*sin(x)+0.15*130000*cos(x) ?

The general solution for 73500=130000*sin(x)+0.15*130000*cos(x) is x=2.39936…+2pin,x=0.44444…+2pin

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73500=130000sin(x)+0.15130000*cos(x)

73500 = 130000 \cdot sin (x) + 0.15 \cdot 130000 \cdot cos (x)

x = 2.39936 \dots + 2 πn, x = 0.44444 \dots + 2 πn

Solution steps

73500 = 130000 sin (x) + 0.15 \cdot 130000 cos (x)

Subtract

0.15130000 cos (x)

from both sides

130000 sin (x) = 73500 - 19500 cos (x)

Square both sides

(130000 sin (x))^{2} = (73500 - 19500 cos (x))^{2}

Subtract

(73500 - 19500 cos (x))^{2}

from both sides

13000 0^{2} sin^{2} (x) - 7350 0^{2} + 2866500000 cos (x) - 380250000 cos^{2} (x) = 0

Rewrite using trig identities

- 7350 0^{2} + (1 - cos^{2} (x)) \cdot 13000 0^{2} + 2866500000 cos (x) - 380250000 cos^{2} (x) = 0

Solve by substitution

cos (x) = \frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )}, cos (x) = - \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )}

cos (x) = \frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )} : x = arccos (\frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn, x = - arccos (\frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn

cos (x) = - \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )} : x = arccos (- \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn, x = 2 π - arccos (- \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn

Combine all the solutions

x = arccos (\frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn, x = - arccos (\frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn, x = arccos (- \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn, x = 2 π - arccos (- \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn

Verify solutions by plugging them into the original equation

x = arccos (\frac{- 2866500000 + 286650000 0 ^{2} - 5 ^{14} \cdot 59833073664 + 13000 0 ^{4} \cdot 4 - 7350 0 ^{2} \cdot 1521000000 + 13000 0 ^{2} \cdot 1521000000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn, x = arccos (- \frac{13000 0 ^{4} \cdot 4 + 286650000 0 ^{2} + 13000 0 ^{2} \cdot 1521000000 - 5 ^{14} \cdot 59833073664 - 7350 0 ^{2} \cdot 1521000000 + 2866500000}{2 ( - 13000 0 ^{2} - 380250000 )}) + 2 πn

Show solutions in decimal form

x = 2.39936 \dots + 2 πn, x = 0.44444 \dots + 2 πn

cos (\frac{π}{2} + x) - sin (\frac{π}{2} + x) = 0

tan (θ + 3 2^{\circ}) = cot (θ - 2 0^{\circ})

sin^{3} (x) = sin^{3} (x) cos (x)

sec^{2} (x) - 1 = tan (x)

tan (\frac{x}{4}) = 1

What is the general solution for 73500=130000*sin(x)+0.15*130000*cos(x) ?
The general solution for 73500=130000*sin(x)+0.15*130000*cos(x) is x=2.39936…+2pin,x=0.44444…+2pin