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))

Solution

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

Solution

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

Degrees

x = 98.71304 \dots^{\circ} + 36 0^{\circ} n, x = - 98.71304 \dots^{\circ} + 36 0^{\circ} n

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

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

(8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 cos (x))^{2} = 12 0^{2}

Expand

(8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 cos (x))^{2} : - 12450 cos (x) + 12514

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

Apply radical rule:

a = a^{\frac{1}{2}}

= ((8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 cos (x))^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 cos (x))^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

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

Refine

= - 12450 cos (x) + 12514

Expand

12 0^{2} : 14400

12 0^{2}

12 0^{2} = 14400

= 14400

- 12450 cos (x) + 12514 = 14400

- 12450 cos (x) + 12514 = 14400

Solve

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

- 12450 cos (x) + 12514 = 14400

Move

12514

to the right side

- 12450 cos (x) + 12514 = 14400

Subtract

12514

from both sides

- 12450 cos (x) + 12514 - 12514 = 14400 - 12514

Simplify

- 12450 cos (x) = 1886

- 12450 cos (x) = 1886

Divide both sides by

- 12450

- 12450 cos (x) = 1886

Divide both sides by

- 12450

\frac{- 12450 cos ( x )}{- 12450} = \frac{1886}{- 12450}

Simplify

\frac{- 12450 cos ( x )}{- 12450} = \frac{1886}{- 12450}

Simplify

\frac{- 12450 cos ( x )}{- 12450} : cos (x)

\frac{- 12450 cos ( x )}{- 12450}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{12450 cos ( x )}{12450}

Divide the numbers:

\frac{12450}{12450} = 1

= cos (x)

Simplify

\frac{1886}{- 12450} : - \frac{943}{6225}

\frac{1886}{- 12450}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{1886}{12450}

Cancel the common factor:

2

= - \frac{943}{6225}

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

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

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

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

Verify Solutions:

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

True

Check the solutions by plugging them into

8 3^{2} + 7 5^{2} - 28375 cos (x) = 120

Remove the ones that don't agree with the equation.

Plug in

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

True

8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 (- \frac{943}{6225}) = 120

8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 (- \frac{943}{6225}) = 120

8 3^{2} + 7 5^{2} - 2 \cdot 83 \cdot 75 (- \frac{943}{6225})

Apply rule

- (- a) = a

= 8 3^{2} + 7 5^{2} + 2 \cdot 83 \cdot 75 \cdot \frac{943}{6225}

2 \cdot 83 \cdot 75 \cdot \frac{943}{6225} = 1886

2 \cdot 83 \cdot 75 \cdot \frac{943}{6225}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{943 \cdot 2 \cdot 83 \cdot 75}{6225}

Multiply the numbers:

943 \cdot 2 \cdot 83 \cdot 75 = 11740350

= \frac{11740350}{6225}

Divide the numbers:

\frac{11740350}{6225} = 1886

= 1886

= 8 3^{2} + 7 5^{2} + 1886

8 3^{2} = 6889

= 6889 + 7 5^{2} + 1886

7 5^{2} = 5625

= 6889 + 5625 + 1886

Add the numbers:

6889 + 5625 + 1886 = 14400

= 14400

Factor the number:

14400 = 12 0^{2}

= 12 0^{2}

Apply radical rule:

12 0^{2} = 120

= 120

120 = 120

True

The solution is

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

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

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

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Frequently Asked Questions (FAQ)

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