What is the value of 4tan((1749pi)/4)+5cot((327pi)/4) ?

The value of 4tan((1749pi)/4)+5cot((327pi)/4) is -1

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4tan((1749pi)/4)+5cot((327pi)/4)

4 tan (\frac{1749 π}{4}) + 5 cot (\frac{327 π}{4})

- 1

Solution steps

4 tan (\frac{1749 π}{4}) + 5 cot (\frac{327 π}{4})

tan (\frac{1749 π}{4}) = tan (\frac{π}{4})

tan (\frac{1749 π}{4})

Rewrite

\frac{1749 π}{4}

π \cdot 437 + \frac{π}{4}

= tan (π 437 + \frac{π}{4})

Apply the periodicity of

tan

tan (x + π \cdot k) = tan (x)

tan (π \cdot 437 + \frac{π}{4}) = tan (\frac{π}{4})

= tan (\frac{π}{4})

= 4 tan (\frac{π}{4}) + 5 cot (\frac{327 π}{4})

cot (\frac{327 π}{4}) = cot (\frac{3 π}{4})

cot (\frac{327 π}{4})

Rewrite

\frac{327 π}{4}

π \cdot 81 + \frac{3 π}{4}

= cot (π 81 + \frac{3 π}{4})

Apply the periodicity of

cot

cot (x + π \cdot k) = cot (x)

cot (π \cdot 81 + \frac{3 π}{4}) = cot (\frac{3 π}{4})

= cot (\frac{3 π}{4})

= 4 tan (\frac{π}{4}) + 5 cot (\frac{3 π}{4})

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:

cot (\frac{3 π}{4}) = - 1

cot (\frac{3 π}{4})

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= - 1

= 4 \cdot 1 + 5 (- 1)

Simplify

= - 1

sin (22. 2^{\circ})

sin (11 0^{\circ}) cos (6 5^{\circ}) - cos (11 0^{\circ}) sin (6 5^{\circ})

2 sec (\frac{5 π}{6})

sec (\frac{1 π}{6})

sin (285 \circ)

What is the value of 4tan((1749pi)/4)+5cot((327pi)/4) ?
The value of 4tan((1749pi)/4)+5cot((327pi)/4) is -1