What is the value of 6cos^2(pi/8)-3 ?

The value of 6cos^2(pi/8)-3 is (3sqrt(2))/2

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Popular Trigonometry >

6cos^2(pi/8)-3

Solution

6 cos^{2} (\frac{π}{8}) - 3

Solution

\frac{3 2}{2}

Decimal

2.12132 \dots

Solution steps

6 cos^{2} (\frac{π}{8}) - 3

Rewrite using trig identities:

cos^{2} (\frac{π}{8}) = \frac{1 + cos ( \frac{π}{4} )}{2}

cos^{2} (\frac{π}{8})

Use the following identity:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

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

= \frac{1 + cos ( 2 \cdot \frac{π}{8} )}{2}

Simplify:

2 \cdot \frac{π}{8} = \frac{π}{4}

2 \cdot \frac{π}{8}

Multiply fractions:

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

= \frac{π 2}{8}

Cancel the common factor:

2

= \frac{π}{4}

= \frac{1 + cos ( \frac{π}{4} )}{2}

= 6 \cdot \frac{1 + cos ( \frac{π}{4} )}{2} - 3

6 \cdot \frac{1 + cos ( \frac{π}{4} )}{2} - 3 = 3 cos (\frac{π}{4})

6 \cdot \frac{1 + cos ( \frac{π}{4} )}{2} - 3

6 \cdot \frac{1 + cos ( \frac{π}{4} )}{2} = 3 (cos (\frac{π}{4}) + 1)

6 \cdot \frac{1 + cos ( \frac{π}{4} )}{2}

Multiply fractions:

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

= \frac{( 1 + cos ( \frac{π}{4} ) ) \cdot 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 (cos (\frac{π}{4}) + 1)

= 3 (cos (\frac{π}{4}) + 1) - 3

Expand

3 (cos (\frac{π}{4}) + 1) : 3 cos (\frac{π}{4}) + 3

3 (cos (\frac{π}{4}) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = cos (\frac{π}{4}), c = 1

= 3 cos (\frac{π}{4}) + 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

= 3 cos (\frac{π}{4}) + 3

= 3 cos (\frac{π}{4}) + 3 - 3

3 - 3 = 0

= 3 cos (\frac{π}{4})

= 3 cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{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}

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

Simplify

3 \cdot \frac{2}{2} : \frac{3 2}{2}

3 \cdot \frac{2}{2}

Multiply fractions:

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

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

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

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

Apply radical rule:

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

= \frac{3}{2}

Rationalize

\frac{3}{2} : \frac{3 2}{2}

\frac{3}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3 2}{2}

= \frac{3 2}{2}

= \frac{3 2}{2}

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Frequently Asked Questions (FAQ)

What is the value of 6cos^2(pi/8)-3 ?
The value of 6cos^2(pi/8)-3 is (3sqrt(2))/2