What is the value of (cot(pi/3)+csc(pi/3))/(cos(pi/4)) ?

The value of (cot(pi/3)+csc(pi/3))/(cos(pi/4)) is square root of 6

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(cot(pi/3)+csc(pi/3))/(cos(pi/4))

Solution

\frac{cot ( \frac{π}{3} ) + csc ( \frac{π}{3} )}{cos ( \frac{π}{4} )}

Solution

6

Decimal Notation

2.44948 \dots

Solution steps

\frac{cot ( \frac{π}{3} ) + csc ( \frac{π}{3} )}{cos ( \frac{π}{4} )}

Use the following trivial identity:

cot (\frac{π}{3}) = \frac{3}{3}

cot (\frac{π}{3})

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

= \frac{3}{3}

Use the following trivial identity:

csc (\frac{π}{3}) = \frac{2 3}{3}

csc (\frac{π}{3})

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= \frac{2 3}{3}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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{2}{2}

= \frac{\frac{3}{3} + \frac{2 3}{3}}{\frac{2}{2}}

Simplify

\frac{\frac{3}{3} + \frac{2 3}{3}}{\frac{2}{2}} : 6

\frac{\frac{3}{3} + \frac{2 3}{3}}{\frac{2}{2}}

Combine the fractions

\frac{3}{3} + \frac{2 3}{3} : 3

Apply rule

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

= \frac{3 + 2 3}{3}

Add similar elements:

3 + 23 = 33

= \frac{3 3}{3}

Divide the numbers:

\frac{3}{3} = 1

= 3

= \frac{3}{\frac{2}{2}}

Apply the fraction rule:

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

= \frac{3 \cdot 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 3}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 3 \cdot 2^{- \frac{1}{2} + 1}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3 \cdot 2^{\frac{1}{2}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= 32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= 6

Popular Examples

cot(-540)

cot (- 54 0^{\circ})

tan(-285)

tan (- 28 5^{\circ})

-3cos^2(pi/2)sin(pi/2)

- 3 cos^{2} (\frac{π}{2}) sin (\frac{π}{2})

csc(480)

csc (48 0^{\circ})

csc(220)

csc (22 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of (cot(pi/3)+csc(pi/3))/(cos(pi/4)) ?
The value of (cot(pi/3)+csc(pi/3))/(cos(pi/4)) is square root of 6