What is the value of sec(30)cos(30)+sin(45)csc(45)-cot(30) ?

The value of sec(30)cos(30)+sin(45)csc(45)-cot(30) is 2-sqrt(3)

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sec(30)cos(30)+sin(45)csc(45)-cot(30)

Solution

sec (3 0^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) csc (4 5^{\circ}) - cot (3 0^{\circ})

Solution

2 - 3

Decimal Notation

0.26794 \dots

Solution steps

sec (3 0^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) csc (4 5^{\circ}) - cot (3 0^{\circ})

Use the following trivial identity:

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

sec (3 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}

= \frac{2 3}{3}

Use the following trivial identity:

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

cos (3 0^{\circ})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\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{2}{2}

Use the following trivial identity:

csc (4 5^{\circ}) = 2

csc (4 5^{\circ})

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

= 2

Use the following trivial identity:

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

cot (3 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

= 3

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= \frac{3 3}{3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

\frac{2}{2} 2 = 1

\frac{2}{2} 2

Multiply fractions:

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

= \frac{2 2}{2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1 + 1 - 3

Add the numbers:

1 + 1 = 2

= 2 - 3

= 2 - 3

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Frequently Asked Questions (FAQ)

What is the value of sec(30)cos(30)+sin(45)csc(45)-cot(30) ?
The value of sec(30)cos(30)+sin(45)csc(45)-cot(30) is 2-sqrt(3)