What is the value of cot(60)+sin(60)-sec(60) ?

The value of cot(60)+sin(60)-sec(60) is (5sqrt(3))/6-2

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Popular Trigonometry >

cot(60)+sin(60)-sec(60)

Solution

cot (6 0^{\circ}) + sin (6 0^{\circ}) - sec (6 0^{\circ})

Solution

\frac{5 3}{6} - 2

Decimal

- 0.55662 \dots

Solution steps

cot (6 0^{\circ}) + sin (6 0^{\circ}) - sec (6 0^{\circ})

Use the following trivial identity:

cot (6 0^{\circ}) = \frac{3}{3}

cot (6 0^{\circ})

cot (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= \frac{3}{3}

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Use the following trivial identity:

sec (6 0^{\circ}) = 2

sec (6 0^{\circ})

sec (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

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

Simplify

\frac{3}{3} + \frac{3}{2} - 2 : \frac{5 3}{6} - 2

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

Combine the fractions

\frac{3}{3} + \frac{3}{2} : \frac{5}{2 3}

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

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{3}{3} :

multiply the denominator and numerator by

2

\frac{3}{3} = \frac{3 \cdot 2}{3 \cdot 2} = \frac{3 \cdot 2}{6}

For

\frac{3}{2} :

multiply the denominator and numerator by

3

\frac{3}{2} = \frac{3 \cdot 3}{2 \cdot 3} = \frac{3 \cdot 3}{6}

= \frac{3 \cdot 2}{6} + \frac{3 \cdot 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{3 \cdot 2 + 3 \cdot 3}{6}

Add similar elements:

23 + 33 = 53

= \frac{5 3}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{5 3}{2 \cdot 3}

Cancel

\frac{5 3}{2 \cdot 3} : \frac{5}{2 3}

\frac{5 3}{2 \cdot 3}

Apply radical rule:

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

= \frac{5 \cdot 3 ^{\frac{1}{2}}}{2 \cdot 3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{5}{2 \cdot 3 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{5}{2 \cdot 3 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{5}{2 3}

= \frac{5}{2 3}

= \frac{5}{2 3} - 2

\frac{5}{2 3} = \frac{5 3}{6}

\frac{5}{2 3}

Multiply by the conjugate

\frac{3}{3}

= \frac{5 3}{2 3 3}

233 = 6

233

Apply radical rule:

a a = a

33 = 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{5 3}{6}

= \frac{5 3}{6} - 2

= \frac{5 3}{6} - 2

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Frequently Asked Questions (FAQ)

What is the value of cot(60)+sin(60)-sec(60) ?
The value of cot(60)+sin(60)-sec(60) is (5sqrt(3))/6-2