What is the value of 2/5 sin^2(60)+3cos^2(30)-tan^2(45) ?

The value of 2/5 sin^2(60)+3cos^2(30)-tan^2(45) is 31/20

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2/5 sin^2(60)+3cos^2(30)-tan^2(45)

Solution

\frac{2}{5} sin^{2} (6 0^{\circ}) + 3 cos^{2} (3 0^{\circ}) - tan^{2} (4 5^{\circ})

Solution

\frac{31}{20}

Decimal Notation

1.55

Solution steps

\frac{2}{5} sin^{2} (6 0^{\circ}) + 3 cos^{2} (3 0^{\circ}) - tan^{2} (4 5^{\circ})

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}

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:

tan (4 5^{\circ}) = 1

tan (4 5^{\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}

= 1

= \frac{2}{5} (\frac{3}{2})^{2} + 3 (\frac{3}{2})^{2} - 1^{2}

Simplify

\frac{2}{5} (\frac{3}{2})^{2} + 3 (\frac{3}{2})^{2} - 1^{2} : \frac{31}{20}

\frac{2}{5} (\frac{3}{2})^{2} + 3 (\frac{3}{2})^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{2}{5} (\frac{3}{2})^{2} + 3 (\frac{3}{2})^{2} - 1

Add similar elements:

\frac{2}{5} (\frac{3}{2})^{2} + 3 (\frac{3}{2})^{2} = \frac{17}{5} (\frac{3}{2})^{2}

\frac{2}{5} (\frac{3}{2})^{2} + 3 (\frac{3}{2})^{2}

Factor out common term

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

= (\frac{3}{2})^{2} (\frac{2}{5} + 3)

\frac{2}{5} + 3 = \frac{17}{5}

\frac{2}{5} + 3

Convert element to fraction:

3 = \frac{3 \cdot 5}{5}

= \frac{2}{5} + \frac{3 \cdot 5}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 3 \cdot 5}{5}

2 + 3 \cdot 5 = 17

2 + 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 2 + 15

Add the numbers:

2 + 15 = 17

= 17

= \frac{17}{5}

= \frac{17}{5} (\frac{3}{2})^{2}

= \frac{17}{5} (\frac{3}{2})^{2} - 1

\frac{17}{5} (\frac{3}{2})^{2} = \frac{51}{20}

\frac{17}{5} (\frac{3}{2})^{2}

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

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

Apply exponent rule:

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

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

= \frac{17}{5} \cdot \frac{3}{2 ^{2}}

Multiply fractions:

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

= \frac{17 \cdot 3}{5 \cdot 2 ^{2}}

Multiply the numbers:

17 \cdot 3 = 51

= \frac{51}{2 ^{2} \cdot 5}

5 \cdot 2^{2} = 20

5 \cdot 2^{2}

2^{2} = 4

= 5 \cdot 4

Multiply the numbers:

5 \cdot 4 = 20

= 20

= \frac{51}{20}

= \frac{51}{20} - 1

Convert element to fraction:

1 = \frac{1 \cdot 20}{20}

= - \frac{1 \cdot 20}{20} + \frac{51}{20}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 20 + 51}{20}

- 1 \cdot 20 + 51 = 31

- 1 \cdot 20 + 51

Multiply the numbers:

1 \cdot 20 = 20

= - 20 + 51

Add/Subtract the numbers:

- 20 + 51 = 31

= 31

= \frac{31}{20}

= \frac{31}{20}

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Frequently Asked Questions (FAQ)

What is the value of 2/5 sin^2(60)+3cos^2(30)-tan^2(45) ?
The value of 2/5 sin^2(60)+3cos^2(30)-tan^2(45) is 31/20