What is the value of 2ln(sec(pi/4)) ?

The value of 2ln(sec(pi/4)) is ln(2)

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2ln(sec(pi/4))

2 ln (sec (\frac{π}{4}))

ln (2)

Decimal Notation

0.69314 \dots

Solution steps

2 ln (sec (\frac{π}{4}))

Use the following trivial identity:

sec (\frac{π}{4}) = 2

sec (\frac{π}{4})

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

= 2 ln (2)

Simplify

2 ln (2) : ln (2)

2 ln (2)

Simplify

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

ln (2)

Rewrite as

= ln (2^{\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 (2)

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

Multiply fractions:

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

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

Cancel the common factor:

2

= ln (2) \cdot 1

Multiply:

ln (2) \cdot 1 = ln (2)

= ln (2)

= ln (2)

2 cos (\frac{π}{6}) + sin (2) (\frac{π}{6})

cos (- 0)

- sin (\frac{π}{6}) + cos (\frac{π}{6})

arccos (0.01)

4 arccos (\frac{0}{2})