What is the value of 3/4 (sin((15pi)/2))+2 ?

The value of 3/4 (sin((15pi)/2))+2 is 5/4

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Popular >

3/4 (sin((15pi)/2))+2

Solution

\frac{3}{4} (sin (\frac{15 π}{2})) + 2

Solution

\frac{5}{4}

Decimal Notation

1.25

Solution steps

\frac{3}{4} (sin (\frac{15 π}{2})) + 2

= \frac{3}{4} sin (\frac{15 π}{2}) + 2

\frac{3}{4} sin (\frac{15 π}{2}) + 2 = \frac{3 sin ( \frac{15 π}{2} ) + 8}{4}

\frac{3}{4} sin (\frac{15 π}{2}) + 2

Multiply

\frac{3}{4} sin (\frac{15 π}{2}) : \frac{3 sin ( \frac{15 π}{2} )}{4}

\frac{3}{4} sin (\frac{15 π}{2})

Multiply fractions:

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

= \frac{3 sin ( \frac{15 π}{2} )}{4}

= \frac{3 sin ( \frac{15 π}{2} )}{4} + 2

Convert element to fraction:

2 = \frac{2 \cdot 4}{4}

= \frac{3 sin ( \frac{15 π}{2} )}{4} + \frac{2 \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{3 sin ( \frac{15 π}{2} ) + 2 \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 sin ( \frac{15 π}{2} ) + 8}{4}

= \frac{3 sin ( \frac{15 π}{2} ) + 8}{4}

sin (\frac{15 π}{2}) = sin (\frac{3 π}{2})

sin (\frac{15 π}{2})

Rewrite

\frac{15 π}{2}

2 π \cdot 3 + \frac{3 π}{2}

= sin (2 π 3 + \frac{3 π}{2})

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

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

= sin (\frac{3 π}{2})

= \frac{3 sin ( \frac{3 π}{2} ) + 8}{4}

Rewrite using trig identities:

sin (\frac{3 π}{2}) = 2 sin (\frac{3 π}{4}) cos (\frac{3 π}{4})

sin (\frac{3 π}{2})

Write

sin (\frac{3 π}{2})

sin (2 \cdot \frac{3 π}{4})

= sin (2 \cdot \frac{3 π}{4})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (\frac{3 π}{4}) cos (\frac{3 π}{4})

= \frac{3 \cdot 2 sin ( \frac{3 π}{4} ) cos ( \frac{3 π}{4} ) + 8}{4}

Simplify

= \frac{6 sin ( \frac{3 π}{4} ) cos ( \frac{3 π}{4} ) + 8}{4}

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

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{2}{2}

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

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{2}{2}

= \frac{6 \cdot \frac{2}{2} ( - \frac{2}{2} ) + 8}{4}

Simplify

\frac{6 \cdot \frac{2}{2} ( - \frac{2}{2} ) + 8}{4} : \frac{5}{4}

\frac{6 \cdot \frac{2}{2} ( - \frac{2}{2} ) + 8}{4}

Remove parentheses:

(- a) = - a

= \frac{- 6 \cdot \frac{2}{2} \cdot \frac{2}{2} + 8}{4}

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

6 \cdot \frac{2}{2} \cdot \frac{2}{2}

Multiply fractions:

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

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

22 \cdot 6 = 12

22 \cdot 6

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 6

Multiply the numbers:

2 \cdot 6 = 12

= 12

= \frac{12}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3

= \frac{- 3 + 8}{4}

Add/Subtract the numbers:

- 3 + 8 = 5

= \frac{5}{4}

= \frac{5}{4}

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Frequently Asked Questions (FAQ)

What is the value of 3/4 (sin((15pi)/2))+2 ?
The value of 3/4 (sin((15pi)/2))+2 is 5/4