What is the general solution for 5sin(x)-5cos(x)=2 ?

The general solution for 5sin(x)-5cos(x)=2 is x=-2.64295…+2pin,x=1.07215…+2pin

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

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

x = - 2.64295 \dots + 2 πn, x = 1.07215 \dots + 2 πn

Solution steps

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

Add

5 cos (x)

to both sides

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

Square both sides

(5 sin (x))^{2} = (2 + 5 cos (x))^{2}

Subtract

(2 + 5 cos (x))^{2}

from both sides

25 sin^{2} (x) - 4 - 20 cos (x) - 25 cos^{2} (x) = 0

Rewrite using trig identities

21 - 20 cos (x) - 50 cos^{2} (x) = 0

Solve by substitution

cos (x) = - \frac{2 + 46}{10}, cos (x) = \frac{46 - 2}{10}

cos (x) = - \frac{2 + 46}{10} : x = arccos (- \frac{2 + 46}{10}) + 2 πn, x = - arccos (- \frac{2 + 46}{10}) + 2 πn

cos (x) = \frac{46 - 2}{10} : x = arccos (\frac{46 - 2}{10}) + 2 πn, x = 2 π - arccos (\frac{46 - 2}{10}) + 2 πn

Combine all the solutions

x = arccos (- \frac{2 + 46}{10}) + 2 πn, x = - arccos (- \frac{2 + 46}{10}) + 2 πn, x = arccos (\frac{46 - 2}{10}) + 2 πn, x = 2 π - arccos (\frac{46 - 2}{10}) + 2 πn

Verify solutions by plugging them into the original equation

x = - arccos (- \frac{2 + 46}{10}) + 2 πn, x = arccos (\frac{46 - 2}{10}) + 2 πn

Show solutions in decimal form

x = - 2.64295 \dots + 2 πn, x = 1.07215 \dots + 2 πn

6 cos^{2} (x) - cos (x) - 2 = 0

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

3 sec (θ) + 2 = 8

sin^{2} (x) = 0.75

- 3 tan (θ) = 3

What is the general solution for 5sin(x)-5cos(x)=2 ?
The general solution for 5sin(x)-5cos(x)=2 is x=-2.64295…+2pin,x=1.07215…+2pin