What is the value of sin(-345)sin(-15) ?

The value of sin(-345)sin(-15) is (sqrt(3)-2)/4

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sin(-345)sin(-15)

Solution

sin (- 34 5^{\circ}) sin (- 1 5^{\circ})

Solution

\frac{3 - 2}{4}

Decimal Notation

- 0.06698 \dots

Solution steps

sin (- 34 5^{\circ}) sin (- 1 5^{\circ})

Rewrite using trig identities:

\frac{cos ( 33 0 ^{\circ} ) - cos ( 0 )}{2}

sin (- 34 5^{\circ}) sin (- 1 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 (- 34 5^{\circ} - (- 1 5^{\circ})) - cos (- 34 5^{\circ} - 1 5^{\circ}))

Simplify

= \frac{cos ( - 33 0 ^{\circ} ) - cos ( - 36 0 ^{\circ} )}{2}

Use the following property:

cos (- x) = cos (x)

cos (- 36 0^{\circ}) = cos (36 0^{\circ})

= \frac{cos ( - 33 0 ^{\circ} ) - cos ( 36 0 ^{\circ} )}{2}

Use the following property:

cos (- x) = cos (x)

cos (- 33 0^{\circ}) = cos (33 0^{\circ})

= \frac{cos ( 33 0 ^{\circ} ) - cos ( 36 0 ^{\circ} )}{2}

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

cos (36 0^{\circ})

Rewrite

36 0^{\circ}

36 0^{\circ} + 0

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

Apply the periodicity of

cos

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

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

= cos (0)

= \frac{cos ( 33 0 ^{\circ} ) - cos ( 0 )}{2}

= \frac{cos ( 33 0 ^{\circ} ) - cos ( 0 )}{2}

Rewrite using trig identities:

cos (33 0^{\circ}) = \frac{3}{2}

cos (33 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (15 0^{\circ}) - sin (18 0^{\circ}) sin (15 0^{\circ})

cos (33 0^{\circ})

Write

cos (33 0^{\circ})

cos (18 0^{\circ} + 15 0^{\circ})

= cos (18 0^{\circ} + 15 0^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (18 0^{\circ}) cos (15 0^{\circ}) - sin (18 0^{\circ}) sin (15 0^{\circ})

= cos (18 0^{\circ}) cos (15 0^{\circ}) - sin (18 0^{\circ}) sin (15 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 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}

= (- 1)

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}

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 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}

= 0

Use the following trivial identity:

sin (15 0^{\circ}) = \frac{1}{2}

sin (15 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}

= (- 1) (- \frac{3}{2}) - 0 \cdot \frac{1}{2}

Simplify

= \frac{3}{2}

Use the following trivial identity:

cos (0) = 1

cos (0)

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}

= 1

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

Simplify

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

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

Join

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

\frac{3}{2} - 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{3 - 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 - 2}{4}

= \frac{3 - 2}{4}

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Frequently Asked Questions (FAQ)

What is the value of sin(-345)sin(-15) ?
The value of sin(-345)sin(-15) is (sqrt(3)-2)/4