What is the general solution for 8^2=14^2+19^2-2(14)(19)cos(C) ?

The general solution for 8^2=14^2+19^2-2(14)(19)cos(C) is C=0.38528…+2pin,C=2pi-0.38528…+2pin

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8^2=14^2+19^2-2(14)(19)cos(C)

8^{2} = 1 4^{2} + 1 9^{2} - 2 (14) (19) cos (C)

C = 0.38528 \dots + 2 πn, C = 2 π - 0.38528 \dots + 2 πn

Degrees

C = 22.07514 \dots^{\circ} + 36 0^{\circ} n, C = 337.92485 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8^{2} = 1 4^{2} + 1 9^{2} - 2 (14) (19) cos (C)

Switch sides

1 4^{2} + 1 9^{2} - 2 \cdot 14 \cdot 19 cos (C) = 8^{2}

Multiply the numbers:

2 \cdot 14 \cdot 19 = 532

1 4^{2} + 1 9^{2} - 532 cos (C) = 8^{2}

1 4^{2} = 196

196 + 1 9^{2} - 532 cos (C) = 8^{2}

1 9^{2} = 361

196 + 361 - 532 cos (C) = 8^{2}

Add the numbers:

196 + 361 = 557

- 532 cos (C) + 557 = 8^{2}

8^{2} = 64

- 532 cos (C) + 557 = 64

Move

557

to the right side

- 532 cos (C) + 557 = 64

Subtract

557

from both sides

- 532 cos (C) + 557 - 557 = 64 - 557

Simplify

- 532 cos (C) = - 493

- 532 cos (C) = - 493

Divide both sides by

- 532

- 532 cos (C) = - 493

Divide both sides by

- 532

\frac{- 532 cos ( C )}{- 532} = \frac{- 493}{- 532}

Simplify

cos (C) = \frac{493}{532}

cos (C) = \frac{493}{532}

Apply trig inverse properties

cos (C) = \frac{493}{532}

General solutions for

cos (C) = \frac{493}{532}

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

C = arccos (\frac{493}{532}) + 2 πn, C = 2 π - arccos (\frac{493}{532}) + 2 πn

C = arccos (\frac{493}{532}) + 2 πn, C = 2 π - arccos (\frac{493}{532}) + 2 πn

Show solutions in decimal form

C = 0.38528 \dots + 2 πn, C = 2 π - 0.38528 \dots + 2 πn

2 sin^{2} (x) = 3 cos (x) + 3

3 tan (x) + 2 = - 3

sec (2 θ) = csc (7 0^{\circ})

- 4 cos^{2} (θ) + 1 = 0

sin (2 x) - sin (4 x) + sin (6 x) = 0

What is the general solution for 8^2=14^2+19^2-2(14)(19)cos(C) ?
The general solution for 8^2=14^2+19^2-2(14)(19)cos(C) is C=0.38528…+2pin,C=2pi-0.38528…+2pin