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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(210)

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

\frac{3}{4}

Decimal Notation

0.75

Solution steps

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

Rewrite using trig identities:

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

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

Use the following identity:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

cos^{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}

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

Simplify

\frac{1 + \frac{1}{2}}{2} : \frac{3}{4}

\frac{1 + \frac{1}{2}}{2}

Join

1 + \frac{1}{2} : \frac{3}{2}

1 + \frac{1}{2}

Convert element to fraction:

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

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

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1

Add the numbers:

2 + 1 = 3

= 3

= \frac{3}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

= \frac{3}{4}

sin (\frac{π}{4}) sin (\frac{π}{12})

cos (12 3^{\circ})

cos (arccos (- 0.1))

7 \cdot sin (5 0^{\circ})

4 sinh (0)