What is the value of ln(cos(pi/6)) ?

The value of ln(cos(pi/6)) is 1/2 ln(3)-ln(2)

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ln(cos(pi/6))

ln (cos (\frac{π}{6}))

\frac{1}{2} ln (3) - ln (2)

Decimal Notation

- 0.14384 \dots

Solution steps

ln (cos (\frac{π}{6}))

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

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{3}{2}

= ln (\frac{3}{2})

Simplify

ln (\frac{3}{2}) : \frac{1}{2} ln (3) - ln (2)

ln (\frac{3}{2})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{3}{2}) = ln (3) - ln (2)

= ln (3) - ln (2)

Simplify

ln (3) : \frac{1}{2} ln (3)

ln (3)

Rewrite as

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

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

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

= \frac{1}{2} ln (3) - ln (2)

= \frac{1}{2} ln (3) - ln (2)

\frac{sin ( 4 5 ^{\circ} )}{2}

\frac{119}{sin ( 2.8 )}

arccot (\frac{7}{24})

4 \cdot cos (0)

cos (\frac{π}{4}) + 4 sin (\frac{π}{4})

What is the value of ln(cos(pi/6)) ?
The value of ln(cos(pi/6)) is 1/2 ln(3)-ln(2)