What is the general solution for 120=sqrt(83^2+75^2-2*83*75*cos(x)) ?

The general solution for 120=sqrt(83^2+75^2-2*83*75*cos(x)) is x=1.72286…+2pin,x=-1.72286…+2pin

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120=sqrt(83^2+75^2-28375*cos(x))

120 = 8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 \cdot cos (x)

x = 1.72286 \dots + 2 πn, x = - 1.72286 \dots + 2 πn

Solution steps

120 = 8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 cos (x)

Switch sides

8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 cos (x) = 120

Square both sides:

- 12450 cos (x) + 12514 = 14400

- 12450 cos (x) + 12514 = 14400

Solve

- 12450 cos (x) + 12514 = 14400 : cos (x) = - \frac{943}{6225}

cos (x) = - \frac{943}{6225}

Verify Solutions:

cos (x) = - \frac{943}{6225}

True

The solution is

cos (x) = - \frac{943}{6225}

Apply trig inverse properties

x = arccos (- \frac{943}{6225}) + 2 πn, x = - arccos (- \frac{943}{6225}) + 2 πn

Show solutions in decimal form

x = 1.72286 \dots + 2 πn, x = - 1.72286 \dots + 2 πn

cos (u) = \frac{7}{25}

sin (x) + cos (x) = 1, 0 \leq x < 2 π

x, cos^{2} (x + y) = sin^{2} (x + y)

cot^{2} (x) + 3 cot (x) + 2 = cot (x) + 1

a^{2} = b^{2} + c^{2} - 2 b c \cdot cos (x)

What is the general solution for 120=sqrt(83^2+75^2-2*83*75*cos(x)) ?
The general solution for 120=sqrt(83^2+75^2-2*83*75*cos(x)) is x=1.72286…+2pin,x=-1.72286…+2pin