What is the value of sin(105)sin(45) ?

The value of sin(105)sin(45) is (1+sqrt(3))/4

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(105)sin(45)

sin (10 5^{\circ}) sin (4 5^{\circ})

\frac{1 + 3}{4}

Decimal Notation

0.68301 \dots

Solution steps

sin (10 5^{\circ}) sin (4 5^{\circ})

Rewrite using trig identities:

\frac{cos ( 6 0 ^{\circ} ) - cos ( 15 0 ^{\circ} )}{2}

sin (10 5^{\circ}) sin (4 5^{\circ})

Use the Product to Sum identity:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= \frac{1}{2} (cos (10 5^{\circ} - 4 5^{\circ}) - cos (10 5^{\circ} + 4 5^{\circ}))

Simplify

= \frac{cos ( 6 0 ^{\circ} ) - cos ( 15 0 ^{\circ} )}{2}

= \frac{cos ( 6 0 ^{\circ} ) - cos ( 15 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}

Use the following trivial identity:

cos (15 0^{\circ}) = - \frac{3}{2}

cos (15 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{\frac{1}{2} - ( - \frac{3}{2} )}{2}

Simplify

\frac{\frac{1}{2} - ( - \frac{3}{2} )}{2} : \frac{1 + 3}{4}

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

Apply rule

- (- a) = a

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

Join

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 3}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1 + 3}{4}

= \frac{1 + 3}{4}

2 (9.8) (0.725) (sin (4 5^{\circ}) 0.12 cos (4 5^{\circ}))

sin (2 \cdot 5 5^{\circ})

180 cos (6 0^{\circ})

cos (\frac{1}{1})

tan (4 5^{\circ}) + sec (6 0^{\circ})

What is the value of sin(105)sin(45) ?
The value of sin(105)sin(45) is (1+sqrt(3))/4