What is the value of 500sec^2(arctan(1/2))(12pi) ?

The value of 500sec^2(arctan(1/2))(12pi) is 7500pi

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500sec^2(arctan(1/2))(12pi)

500 sec^{2} (arctan (\frac{1}{2})) (12 π)

7500 π

Decimal Notation

23561.94490 \dots

Solution steps

500 sec^{2} (arctan (\frac{1}{2})) (12 π)

= 500 sec^{2} (arctan (\frac{1}{2})) \cdot 12 π

Rewrite using trig identities:

sec (arctan (\frac{1}{2})) = \frac{5}{2}

sec (arctan (\frac{1}{2}))

Rewrite using trig identities:

sec (arctan (\frac{1}{2})) = 1 + (\frac{1}{2})^{2}

Use the following identity:

sec (arctan (x)) = 1 + x^{2}

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

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

Simplify

= \frac{5}{2}

= 500 (\frac{5}{2})^{2} \cdot 12 π

Simplify

500 (\frac{5}{2})^{2} \cdot 12 π : 7500 π

500 (\frac{5}{2})^{2} \cdot 12 π

(\frac{5}{2})^{2} = \frac{5}{2 ^{2}}

(\frac{5}{2})^{2}

Apply exponent rule:

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

= \frac{( 5 ) ^{2}}{2 ^{2}}

(5)^{2} : 5

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 5

= \frac{5}{2 ^{2}}

= 500 \cdot 12 π \frac{5}{2 ^{2}}

Multiply fractions:

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

= \frac{5 \cdot 500 \cdot 12 π}{2 ^{2}}

Multiply the numbers:

5 \cdot 500 \cdot 12 = 30000

= \frac{30000 π}{2 ^{2}}

Factor

30000 : 2^{4} \cdot 5^{4} \cdot 3

Factor

30000 = 2^{4} \cdot 5^{4} \cdot 3

= \frac{2 ^{4} \cdot 5 ^{4} \cdot 3 π}{2 ^{2}}

Cancel

\frac{2 ^{4} \cdot 3 \cdot 5 ^{4} π}{2 ^{2}} : 5^{4} \cdot 2^{2} \cdot 3 π

\frac{2 ^{4} \cdot 3 \cdot 5 ^{4} π}{2 ^{2}}

Apply exponent rule:

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

\frac{2 ^{4}}{2 ^{2}} = 2^{4 - 2}

= 5^{4} \cdot 3 \cdot 2^{4 - 2} π

Subtract the numbers:

4 - 2 = 2

= 5^{4} \cdot 2^{2} \cdot 3 π

= 5^{4} \cdot 2^{2} \cdot 3 π

5^{4} = 625

= 2^{2} \cdot 625 \cdot 3 π

2^{2} = 4

= 625 \cdot 4 \cdot 3 π

Multiply the numbers:

625 \cdot 4 \cdot 3 = 7500

= 7500 π

= 7500 π

sin (arccos (\frac{5}{13}))

cos (\frac{5 π}{12}) cos (\frac{π}{4}) + sin (\frac{5 π}{12}) sin (\frac{π}{4})

sec (- \frac{11 π}{6})

arctan (\frac{13}{9})

\frac{1}{sin ( \frac{π}{2} )}

What is the value of 500sec^2(arctan(1/2))(12pi) ?
The value of 500sec^2(arctan(1/2))(12pi) is 7500pi