What is the value of (tan^2(30)sec(45))/(1/(sin^2(60))) ?

The value of (tan^2(30)sec(45))/(1/(sin^2(60))) is (sqrt(2))/4

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(tan^2(30)sec(45))/(1/(sin^2(60)))

Solution

\frac{tan ^{2} ( 3 0 ^{\circ} ) sec ( 4 5 ^{\circ} )}{\frac{1}{s i n ^{2} ( 6 0 ^{\circ} )}}

Solution

\frac{2}{4}

Decimal Notation

0.35355 \dots

Solution steps

\frac{tan ^{2} ( 3 0 ^{\circ} ) sec ( 4 5 ^{\circ} )}{\frac{1}{s i n ^{2} ( 6 0 ^{\circ} )}}

Use the following trivial identity:

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

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

= \frac{3}{3}

Use the following trivial identity:

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

sec (4 5^{\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

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}

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

Simplify

\frac{( \frac{3}{3} ) ^{2} 2}{\frac{1}{( \frac{3}{2} ) ^{2}}} : \frac{2}{4}

\frac{( \frac{3}{3} ) ^{2} 2}{\frac{1}{( \frac{3}{2} ) ^{2}}}

Apply the fraction rule:

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

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

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

(\frac{3}{3})^{2} 2 (\frac{3}{2})^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(\frac{3}{3})^{2} (\frac{3}{2})^{2} = (\frac{3}{3} \cdot \frac{3}{2})^{2}

= 2 (\frac{3}{3} \cdot \frac{3}{2})^{2}

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

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

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{1}{3}

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

Cross-cancel:

3

= \frac{1}{2}

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

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

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

(\frac{1}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply

\frac{1}{2 ^{2}} 2 : \frac{1}{2 2}

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

Multiply fractions:

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

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

Multiply:

1 \cdot 2 = 2

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= \frac{1}{2 2}

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

Apply the fraction rule:

\frac{a}{1} = a

= \frac{1}{2 2}

Rationalize

\frac{1}{2 2} : \frac{2}{4}

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4}

= \frac{2}{4}

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Frequently Asked Questions (FAQ)

What is the value of (tan^2(30)sec(45))/(1/(sin^2(60))) ?
The value of (tan^2(30)sec(45))/(1/(sin^2(60))) is (sqrt(2))/4