What is the value of tan((57pi)/4)+sec((61pi)/3) ?

The value of tan((57pi)/4)+sec((61pi)/3) is 3

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tan((57pi)/4)+sec((61pi)/3)

tan (\frac{57 π}{4}) + sec (\frac{61 π}{3})

3

Solution steps

tan (\frac{57 π}{4}) + sec (\frac{61 π}{3})

tan (\frac{57 π}{4}) = tan (\frac{π}{4})

tan (\frac{57 π}{4})

Rewrite

\frac{57 π}{4}

π \cdot 14 + \frac{π}{4}

= tan (π 14 + \frac{π}{4})

Apply the periodicity of

tan

tan (x + π \cdot k) = tan (x)

tan (π \cdot 14 + \frac{π}{4}) = tan (\frac{π}{4})

= tan (\frac{π}{4})

= tan (\frac{π}{4}) + sec (\frac{61 π}{3})

sec (\frac{61 π}{3}) = sec (\frac{π}{3})

sec (\frac{61 π}{3})

Rewrite

\frac{61 π}{3}

2 π \cdot 10 + \frac{π}{3}

= sec (2 π 10 + \frac{π}{3})

Apply the periodicity of

sec

sec (x + 2 π \cdot k) = sec (x)

sec (2 π \cdot 10 + \frac{π}{3}) = sec (\frac{π}{3})

= sec (\frac{π}{3})

= tan (\frac{π}{4}) + sec (\frac{π}{3})

Use the following trivial identity:

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

Use the following trivial identity:

sec (\frac{π}{3}) = 2

sec (\frac{π}{3})

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

= 1 + 2

Simplify

= 3

1 - cos (3 2^{\circ})

- 6 sin (\frac{2 π}{3})

\frac{53 sin ( 6 )}{sin ( 11 6 ^{\circ} )}

π + arctan (\frac{1}{3})

- sin (\frac{2 π}{4})

What is the value of tan((57pi)/4)+sec((61pi)/3) ?
The value of tan((57pi)/4)+sec((61pi)/3) is 3