What is the value of 20sin(pi/2+pi/3) ?

The value of 20sin(pi/2+pi/3) is 10

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20sin(pi/2+pi/3)

Solution

20 sin (\frac{π}{2} + \frac{π}{3})

Solution

10

Solution steps

20 sin (\frac{π}{2} + \frac{π}{3})

Rewrite using trig identities:

sin (\frac{π}{2} + \frac{π}{3}) = \frac{1}{2}

sin (\frac{π}{2} + \frac{π}{3})

Simplify:

\frac{π}{2} + \frac{π}{3} = \frac{5 π}{6}

\frac{π}{2} + \frac{π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π 3}{6} + \frac{π 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π 2}{6}

Add similar elements:

3 π + 2 π = 5 π

= \frac{5 π}{6}

= sin (\frac{5 π}{6})

Use the following trivial identity:

sin (\frac{5 π}{6}) = \frac{1}{2}

sin (\frac{5 π}{6})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{1}{2}

= 20 \cdot \frac{1}{2}

Simplify

20 \cdot \frac{1}{2} : 10

20 \cdot \frac{1}{2}

Convert element to fraction:

20 = \frac{20}{1}

= \frac{1}{2} \cdot \frac{20}{1}

Cross-cancel common factor:

2

= \frac{10}{1}

Apply the fraction rule:

\frac{a}{1} = a

= 10

= 10

Popular Examples

20cos(120)

20 cos (12 0^{\circ})

arccos((-8)/(12.36))

arccos (\frac{- 8}{12.36})

csc(-1/2)

csc (- \frac{1}{2})

-3cos(pi/3)-2sin(pi/3)

- 3 cos (\frac{π}{3}) - 2 sin (\frac{π}{3})

arcsec(3/4)

arcsec (\frac{3}{4})

Frequently Asked Questions (FAQ)

What is the value of 20sin(pi/2+pi/3) ?
The value of 20sin(pi/2+pi/3) is 10