What is the value of sec^2(pi/4)+4csc^2(pi/3) ?

The value of sec^2(pi/4)+4csc^2(pi/3) is 22/3

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Popular >

sec^2(pi/4)+4csc^2(pi/3)

Solution

sec^{2} (\frac{π}{4}) + 4 csc^{2} (\frac{π}{3})

Solution

\frac{22}{3}

Decimal Notation

7.33333 \dots

Solution steps

sec^{2} (\frac{π}{4}) + 4 csc^{2} (\frac{π}{3})

Use the following trivial identity:

sec (\frac{π}{4}) = 2

sec (\frac{π}{4})

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

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}

= (2)^{2} + 4 (\frac{2 3}{3})^{2}

Simplify

(2)^{2} + 4 (\frac{2 3}{3})^{2} : \frac{22}{3}

(2)^{2} + 4 (\frac{2 3}{3})^{2}

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

4 (\frac{2 3}{3})^{2} = \frac{16}{3}

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(23)^{2} = 2^{2} (3)^{2}

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

(3)^{2} : 3

Apply radical rule:

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

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

Apply exponent rule:

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

= 3^{\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

= 3

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

Cancel the common factor:

3

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

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

Multiply fractions:

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

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

2^{2} \cdot 4 = 16

2^{2} \cdot 4

2^{2} = 4

= 4 \cdot 4

Multiply the numbers:

4 \cdot 4 = 16

= 16

= \frac{16}{3}

= 2 + \frac{16}{3}

Convert element to fraction:

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

= \frac{2 \cdot 3}{3} + \frac{16}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 3 + 16}{3}

2 \cdot 3 + 16 = 22

2 \cdot 3 + 16

Multiply the numbers:

2 \cdot 3 = 6

= 6 + 16

Add the numbers:

6 + 16 = 22

= 22

= \frac{22}{3}

= \frac{22}{3}

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Frequently Asked Questions (FAQ)

What is the value of sec^2(pi/4)+4csc^2(pi/3) ?
The value of sec^2(pi/4)+4csc^2(pi/3) is 22/3