What is the general solution for 7.9615=59.2153cos(x)-12sin(x) ?

The general solution for 7.9615=59.2153cos(x)-12sin(x) is x=pi+1.50300…+2pin,x=1.23869…+2pin

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7.9615=59.2153cos(x)-12sin(x)

7.9615 = 59.2153 cos (x) - 12 sin (x)

x = π + 1.50300 \dots + 2 πn, x = 1.23869 \dots + 2 πn

Solution steps

7.9615 = 59.2153 cos (x) - 12 sin (x)

Add

12 sin (x)

to both sides

59.2153 cos (x) = 7.9615 + 12 sin (x)

Square both sides

(59.2153 cos (x))^{2} = (7.9615 + 12 sin (x))^{2}

Subtract

(7.9615 + 12 sin (x))^{2}

from both sides

3506.45175 \dots cos^{2} (x) - 63.38548 \dots - 191.076 sin (x) - 144 sin^{2} (x) = 0

Rewrite using trig identities

3443.06627 \dots - 191.076 sin (x) - 3650.45175 \dots sin^{2} (x) = 0

Solve by substitution

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}, sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots} : x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn

sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots} : x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn, x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn, x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Verify solutions by plugging them into the original equation

x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Show solutions in decimal form

x = π + 1.50300 \dots + 2 πn, x = 1.23869 \dots + 2 πn

1 - tan (x) = 0

2 cos^{2} (x) - 3 cos (x) = 0, 0^{\circ} < x < 18 0^{\circ}

\frac{tan ( x )}{cos ( x )} = \frac{2}{3}

tan (x) + 2 cos (x) csc (x) = sec (x) cos (x) + cot (x)

cos (x) = 1.22719

What is the general solution for 7.9615=59.2153cos(x)-12sin(x) ?
The general solution for 7.9615=59.2153cos(x)-12sin(x) is x=pi+1.50300…+2pin,x=1.23869…+2pin