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

The value of ln(sec(pi/4)) is 1/2 ln(2)

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ln(sec(pi/4))

ln (sec (\frac{π}{4}))

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

Decimal Notation

0.34657 \dots

Solution steps

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

= 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)

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

arctan (\frac{5}{11})

- 7 cos (2)

arctan (\frac{5}{13})

sin (- 0)

\frac{cos ^{3} ( π )}{3}