What is the value of 4cos^2(60)+4sin^2(45)-sin^2(30) ?

The value of 4cos^2(60)+4sin^2(45)-sin^2(30) is 11/4

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4cos^2(60)+4sin^2(45)-sin^2(30)

Solution

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

Solution

\frac{11}{4}

Decimal Notation

2.75

Solution steps

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

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 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{1}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\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{2}{2}

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

= 4 (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2} - (\frac{1}{2})^{2}

Simplify

4 (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2} - (\frac{1}{2})^{2} : \frac{11}{4}

4 (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2} - (\frac{1}{2})^{2}

Group like terms

= 4 (\frac{1}{2})^{2} - (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2}

Add similar elements:

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

= 3 (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2}

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

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

(\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}}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

2^{2} = 4

= \frac{3}{4}

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

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

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

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

Apply exponent rule:

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

= \frac{( 2 ) ^{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

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

Cancel the common factor:

2

= \frac{1}{2}

= 4 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= \frac{3}{4} + 2

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 4 + 3 = 11

2 \cdot 4 + 3

Multiply the numbers:

2 \cdot 4 = 8

= 8 + 3

Add the numbers:

8 + 3 = 11

= 11

= \frac{11}{4}

= \frac{11}{4}

Popular Examples

sin((3pi)/(18))

sin (\frac{3 π}{18})

arccos(7/15)

arccos (\frac{7}{15})

tan(-(29pi)/3)

tan (- \frac{29 π}{3})

sec((11pi)/6)*tan((11pi)/6)

sec (\frac{11 π}{6}) \cdot tan (\frac{11 π}{6})

arccos(-57.989)

arccos (- 57.989)

Frequently Asked Questions (FAQ)

What is the value of 4cos^2(60)+4sin^2(45)-sin^2(30) ?
The value of 4cos^2(60)+4sin^2(45)-sin^2(30) is 11/4