What is the value of (sec(30)+sin(60))^2 ?

The value of (sec(30)+sin(60))^2 is 49/12

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Popular >

(sec(30)+sin(60))^2

Solution

(sec (3 0^{\circ}) + sin (6 0^{\circ}))^{2}

Solution

\frac{49}{12}

Decimal Notation

4.08333 \dots

Solution steps

(sec (3 0^{\circ}) + sin (6 0^{\circ}))^{2}

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:

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

Simplify

(\frac{2 3}{3} + \frac{3}{2})^{2} : \frac{49}{12}

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

Join

\frac{2 3}{3} + \frac{3}{2} : \frac{7}{2 3}

\frac{2 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{2 3}{3} :

multiply the denominator and numerator by

2

\frac{2 3}{3} = \frac{2 3 \cdot 2}{3 \cdot 2} = \frac{4 3}{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{4 3}{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{4 3 + 3 \cdot 3}{6}

Add similar elements:

43 + 33 = 73

= \frac{7 3}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{7 3}{2 \cdot 3}

Cancel

\frac{7 3}{2 \cdot 3} : \frac{7}{2 3}

\frac{7 3}{2 \cdot 3}

Apply radical rule:

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

= \frac{7 \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{7}{2 \cdot 3 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{7}{2 \cdot 3 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{7}{2 3}

= \frac{7}{2 3}

= (\frac{7}{2 3})^{2}

Apply exponent rule:

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

= \frac{7 ^{2}}{( 2 3 ) ^{2}}

Apply exponent rule:

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

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

= \frac{7 ^{2}}{2 ^{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{7 ^{2}}{2 ^{2} \cdot 3}

7^{2} = 49

= \frac{49}{2 ^{2} \cdot 3}

2^{2} \cdot 3 = 12

2^{2} \cdot 3

2^{2} = 4

= 4 \cdot 3

Multiply the numbers:

4 \cdot 3 = 12

= 12

= \frac{49}{12}

= \frac{49}{12}

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Frequently Asked Questions (FAQ)

What is the value of (sec(30)+sin(60))^2 ?
The value of (sec(30)+sin(60))^2 is 49/12