What is the value of cos^2(135)-sin^2(210) ?

The value of cos^2(135)-sin^2(210) is 1/4

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(135)-sin^2(210)

Solution

cos^{2} (13 5^{\circ}) - sin^{2} (21 0^{\circ})

Solution

\frac{1}{4}

Decimal Notation

0.25

Solution steps

cos^{2} (13 5^{\circ}) - sin^{2} (21 0^{\circ})

Rewrite using trig identities:

sin^{2} (21 0^{\circ}) = \frac{1 - cos ( 6 0 ^{\circ} )}{2}

sin^{2} (21 0^{\circ})

Use the following identity:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

= \frac{1 - cos ( 2 \cdot 21 0 ^{\circ} )}{2}

Simplify

= \frac{1 - cos ( 42 0 ^{\circ} )}{2}

cos (42 0^{\circ}) = cos (6 0^{\circ})

cos (42 0^{\circ})

Rewrite

42 0^{\circ}

36 0^{\circ} + 6 0^{\circ}

= cos (36 0^{\circ} + 6 0^{\circ})

Apply the periodicity of

cos

cos (x + 36 0^{\circ}) = cos (x)

cos (36 0^{\circ} + 6 0^{\circ}) = cos (6 0^{\circ})

= cos (6 0^{\circ})

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

= cos^{2} (13 5^{\circ}) - \frac{1 - cos ( 6 0 ^{\circ} )}{2}

Simplify

= \frac{2 cos ^{2} ( 13 5 ^{\circ} ) - 1 + cos ( 6 0 ^{\circ} )}{2}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\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{2}{2}

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}

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

Simplify

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

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

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

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

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

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

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

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

2 (\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}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

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

1 - 1 = 0

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4}

Popular Examples

sin(arctan(-5/12))

sin (arctan (- \frac{5}{12}))

cos(140)+cos(40)

cos (14 0^{\circ}) + cos (4 0^{\circ})

cos(arccot(3/4))

cos (arccot (\frac{3}{4}))

200sin(45)

200 sin (4 5^{\circ})

7sin(75)

7 sin (7 5^{\circ})

Frequently Asked Questions (FAQ)

What is the value of cos^2(135)-sin^2(210) ?
The value of cos^2(135)-sin^2(210) is 1/4