What is the value of -13.5cos(pi/6*45)+14.5 ?

The value of -13.5cos(pi/6*45)+14.5 is 29/2

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Popular Trigonometry >

-13.5cos(pi/6*45)+14.5

Solution

- 13.5 cos (\frac{π}{6} \cdot 45) + 14.5

Solution

\frac{29}{2}

Decimal

14.5

Solution steps

- 13.5 cos (\frac{π}{6} \cdot 45) + 14.5

= - \frac{27}{2} cos (\frac{π}{6} \cdot 45) + \frac{29}{2}

Simplify:

\frac{π}{6} \cdot 45 = \frac{15 π}{2}

\frac{π}{6} \cdot 45

Multiply fractions:

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

= \frac{π 45}{6}

Cancel the common factor:

3

= \frac{15 π}{2}

- \frac{27}{2} cos (\frac{15 π}{2}) + \frac{29}{2} = \frac{- 27 cos ( \frac{15 π}{2} ) + 29}{2}

- \frac{27}{2} cos (\frac{15 π}{2}) + \frac{29}{2}

Multiply

\frac{27}{2} cos (\frac{15 π}{2}) : \frac{27 cos ( \frac{15 π}{2} )}{2}

\frac{27}{2} cos (\frac{15 π}{2})

Multiply fractions:

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

= \frac{27 cos ( \frac{15 π}{2} )}{2}

= - \frac{27 cos ( \frac{15 π}{2} )}{2} + \frac{29}{2}

Apply rule

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

= \frac{- 27 cos ( \frac{15 π}{2} ) + 29}{2}

= \frac{- 27 cos ( \frac{15 π}{2} ) + 29}{2}

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

cos (\frac{15 π}{2})

Rewrite

\frac{15 π}{2}

2 π \cdot 3 + \frac{3 π}{2}

= cos (2 π 3 + \frac{3 π}{2})

Apply the periodicity of

cos

cos (x + 2 π \cdot k) = cos (x)

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

= cos (\frac{3 π}{2})

= \frac{- 27 cos ( \frac{3 π}{2} ) + 29}{2}

Rewrite using trig identities:

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

cos (\frac{3 π}{2})

Write

cos (\frac{3 π}{2})

cos (2 \cdot \frac{3 π}{4})

= cos (2 \cdot \frac{3 π}{4})

Use the Double Angle identity:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

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

= \frac{- 27 ( cos ^{2} ( \frac{3 π}{4} ) - sin ^{2} ( \frac{3 π}{4} ) ) + 29}{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}

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}

= \frac{- 27 ( ( - \frac{2}{2} ) ^{2} - ( \frac{2}{2} ) ^{2} ) + 29}{2}

Simplify

\frac{- 27 ( ( - \frac{2}{2} ) ^{2} - ( \frac{2}{2} ) ^{2} ) + 29}{2} : \frac{29}{2}

\frac{- 27 ( ( - \frac{2}{2} ) ^{2} - ( \frac{2}{2} ) ^{2} ) + 29}{2}

- 27 (- \frac{2}{2})^{2} - (\frac{2}{2})^{2} + 29 = 29

- 27 (- \frac{2}{2})^{2} - (\frac{2}{2})^{2} + 29

27 (- \frac{2}{2})^{2} - (\frac{2}{2})^{2} = 0

27 (- \frac{2}{2})^{2} - (\frac{2}{2})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{2}{2})^{2} = (\frac{2}{2})^{2}

= 27 (\frac{2}{2})^{2} - (\frac{2}{2})^{2}

Add similar elements:

(\frac{2}{2})^{2} - (\frac{2}{2})^{2} = 0

= 27 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= - 0 + 29

Add/Subtract the numbers:

- 0 + 29 = 29

= 29

= \frac{29}{2}

= \frac{29}{2}

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Frequently Asked Questions (FAQ)

What is the value of -13.5cos(pi/6*45)+14.5 ?
The value of -13.5cos(pi/6*45)+14.5 is 29/2