What is the value of sqrt((100^2+120^2-2*100*120*cos(60))) ?

The value of sqrt((100^2+120^2-2*100*120*cos(60))) is 20sqrt(31)

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sqrt((100^2+120^2-2100120*cos(60)))

(10 0^{2} + 12 0^{2} - 2 \cdot 100 \cdot 120 \cdot cos (6 0^{\circ}))

2031

Decimal

111.35528 \dots

Solution steps

(10 0^{2} + 12 0^{2} - 2 \cdot 100 \cdot 120 cos (6 0^{\circ}))

= 10 0^{2} + 12 0^{2} - 2 \cdot 100 \cdot 120 cos (6 0^{\circ})

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\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{1}{2}

= 10 0^{2} + 12 0^{2} - 2 \cdot 100 \cdot 120 \cdot \frac{1}{2}

Simplify

10 0^{2} + 12 0^{2} - 2 \cdot 100 \cdot 120 \cdot \frac{1}{2} : 2031

10 0^{2} + 12 0^{2} - 2 \cdot 100 \cdot 120 \cdot \frac{1}{2}

2 \cdot 100 \cdot 120 \cdot \frac{1}{2} = 12000

2 \cdot 100 \cdot 120 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2 \cdot 100 \cdot 120}{2}

Cancel the common factor:

2

= 1 \cdot 100 \cdot 120

Multiply the numbers:

1 \cdot 100 \cdot 120 = 12000

= 12000

= 10 0^{2} + 12 0^{2} - 12000

10 0^{2} = 10000

= 10000 + 12 0^{2} - 12000

12 0^{2} = 14400

= 10000 + 14400 - 12000

Add/Subtract the numbers:

10000 + 14400 - 12000 = 12400

= 12400

Prime factorization of

12400 : 2^{4} \cdot 5^{2} \cdot 31

12400

12400

divides by

212400 = 6200 \cdot 2

= 2 \cdot 6200

6200

divides by

26200 = 3100 \cdot 2

= 2 \cdot 2 \cdot 3100

3100

divides by

23100 = 1550 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 1550

1550

divides by

21550 = 775 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 775

775

divides by

5775 = 155 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 155

155

divides by

5155 = 31 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 31

2, 5, 31

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 31

= 2^{4} \cdot 5^{2} \cdot 31

= 2^{4} \cdot 5^{2} \cdot 31

Apply radical rule:

= 31 2^{4} 5^{2}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 31 5^{2}

Apply radical rule:

5^{2} = 5

= 2^{2} \cdot 531

Refine

= 2031

= 2031

What is the value of sqrt((100^2+120^2-2*100*120*cos(60))) ?
The value of sqrt((100^2+120^2-2*100*120*cos(60))) is 20sqrt(31)