What is the value of 720^2+320^2-2*720*320*cos(45) ?

The value of 720^2+320^2-2*720*320*cos(45) is 620800-230400sqrt(2)

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720^2+320^2-2720320*cos(45)

72 0^{2} + 32 0^{2} - 2 \cdot 720 \cdot 320 \cdot cos (4 5^{\circ})

620800 - 2304002

Decimal

294965.19522 \dots

Solution steps

72 0^{2} + 32 0^{2} - 2 \cdot 720 \cdot 320 cos (4 5^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= 72 0^{2} + 32 0^{2} - 2 \cdot 720 \cdot 320 \cdot \frac{2}{2}

Simplify

72 0^{2} + 32 0^{2} - 2 \cdot 720 \cdot 320 \cdot \frac{2}{2} : 620800 - 2304002

72 0^{2} + 32 0^{2} - 2 \cdot 720 \cdot 320 \cdot \frac{2}{2}

2 \cdot 720 \cdot 320 \cdot \frac{2}{2} = 2304002

2 \cdot 720 \cdot 320 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 2 \cdot 720 \cdot 320}{2}

Cancel the common factor:

2

= 2 \cdot 720 \cdot 320

Multiply the numbers:

720 \cdot 320 = 230400

= 2304002

= 72 0^{2} + 32 0^{2} - 2304002

72 0^{2} = 518400

= 518400 + 32 0^{2} - 2304002

32 0^{2} = 102400

= 518400 + 102400 - 2304002

Add the numbers:

518400 + 102400 = 620800

= 620800 - 2304002

= 620800 - 2304002

What is the value of 720^2+320^2-2*720*320*cos(45) ?
The value of 720^2+320^2-2*720*320*cos(45) is 620800-230400sqrt(2)