What is the value of sqrt(3)sin(pi/3)-1/2 cos^2(pi/3)-7/8 ?

The value of sqrt(3)sin(pi/3)-1/2 cos^2(pi/3)-7/8 is 1/2

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sqrt(3)sin(pi/3)-1/2 cos^2(pi/3)-7/8

Solution

3 sin (\frac{π}{3}) - \frac{1}{2} cos^{2} (\frac{π}{3}) - \frac{7}{8}

Solution

\frac{1}{2}

Decimal Notation

0.5

Solution steps

3 sin (\frac{π}{3}) - \frac{1}{2} cos^{2} (\frac{π}{3}) - \frac{7}{8}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

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{3}{2}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} 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}

= 3 \frac{3}{2} - \frac{1}{2} (\frac{1}{2})^{2} - \frac{7}{8}

Simplify

3 \frac{3}{2} - \frac{1}{2} (\frac{1}{2})^{2} - \frac{7}{8} : \frac{1}{2}

3 \frac{3}{2} - \frac{1}{2} (\frac{1}{2})^{2} - \frac{7}{8}

3 \frac{3}{2} = \frac{3}{2}

3 \frac{3}{2}

Multiply fractions:

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

= \frac{3 3}{2}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{2}

\frac{1}{2} (\frac{1}{2})^{2} = \frac{1}{2 ^{3}}

\frac{1}{2} (\frac{1}{2})^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

\frac{1}{2} (\frac{1}{2})^{2} = (\frac{1}{2})^{1 + 2}

= (\frac{1}{2})^{1 + 2}

Add the numbers:

1 + 2 = 3

= (\frac{1}{2})^{3}

Apply exponent rule:

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

= \frac{1 ^{3}}{2 ^{3}}

Apply rule

1^{a} = 1

1^{3} = 1

= \frac{1}{2 ^{3}}

= \frac{3}{2} - \frac{1}{2 ^{3}} - \frac{7}{8}

2^{3} = 8

2^{3}

2^{3} = 8

= 8

= \frac{3}{2} - \frac{1}{8} - \frac{7}{8}

Least Common Multiplier of

2, 8, 8 : 8

2, 8, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Compute a number comprised of factors that appear in at least one of the following:

2, 8, 8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

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

8

For

\frac{3}{2} :

multiply the denominator and numerator by

4

\frac{3}{2} = \frac{3 \cdot 4}{2 \cdot 4} = \frac{12}{8}

= \frac{12}{8} - \frac{1}{8} - \frac{7}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{12 - 1 - 7}{8}

Subtract the numbers:

12 - 1 - 7 = 4

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

= \frac{1}{2}

Popular Examples

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\frac{1}{cos ^{2} ( \frac{π}{3} )}

arccos(5/(sqrt(9)\sqrt{14)})

arccos (\frac{5}{9 14})

Frequently Asked Questions (FAQ)

What is the value of sqrt(3)sin(pi/3)-1/2 cos^2(pi/3)-7/8 ?
The value of sqrt(3)sin(pi/3)-1/2 cos^2(pi/3)-7/8 is 1/2