What is the value of sin^2(45)+8cos^2(30) ?

The value of sin^2(45)+8cos^2(30) is 13/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin^2(45)+8cos^2(30)

Solution

sin^{2} (4 5^{\circ}) + 8 cos^{2} (3 0^{\circ})

Solution

\frac{13}{2}

Decimal Notation

6.5

Solution steps

sin^{2} (4 5^{\circ}) + 8 cos^{2} (3 0^{\circ})

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:

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}

= (\frac{2}{2})^{2} + 8 (\frac{3}{2})^{2}

Simplify

(\frac{2}{2})^{2} + 8 (\frac{3}{2})^{2} : \frac{13}{2}

(\frac{2}{2})^{2} + 8 (\frac{3}{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}

8 (\frac{3}{2})^{2} = 6

8 (\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}}

= 8 \cdot \frac{3}{2 ^{2}}

Multiply fractions:

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

= \frac{3 \cdot 8}{2 ^{2}}

Multiply the numbers:

3 \cdot 8 = 24

= \frac{24}{2 ^{2}}

Factor

24 : 2^{3} \cdot 3

Factor

24 = 2^{3} \cdot 3

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

Cancel

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

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

Apply exponent rule:

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

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

= 3 \cdot 2^{3 - 2}

Subtract the numbers:

3 - 2 = 1

= 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{1}{2} + 6

Convert element to fraction:

6 = \frac{6 \cdot 2}{2}

= \frac{6 \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{6 \cdot 2 + 1}{2}

6 \cdot 2 + 1 = 13

6 \cdot 2 + 1

Multiply the numbers:

6 \cdot 2 = 12

= 12 + 1

Add the numbers:

12 + 1 = 13

= 13

= \frac{13}{2}

= \frac{13}{2}

Popular Examples

cos(0.6593)

cos (0.6593)

tan(30)tan(60)

tan (3 0^{\circ}) tan (6 0^{\circ})

sin(pi/4+pi/4)

sin (\frac{π}{4} + \frac{π}{4})

arccos(30/40)

arccos (\frac{30}{40})

36*sin(60)

36 \cdot sin (6 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of sin^2(45)+8cos^2(30) ?
The value of sin^2(45)+8cos^2(30) is 13/2