What is the value of cos(105)-cos(15) ?

The value of cos(105)-cos(15) is -sqrt(3/2)

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cos(105)-cos(15)

cos (10 5^{\circ}) - cos (1 5^{\circ})

- \frac{3}{2}

Decimal Notation

- 1.22474 \dots

Solution steps

cos (10 5^{\circ}) - cos (1 5^{\circ})

Rewrite using trig identities:

- 2 sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ}) - cos (1 5^{\circ})

Use the Sum to Product identity:

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

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

Simplify

= - 2 sin (6 0^{\circ}) sin (4 5^{\circ})

= - 2 sin (6 0^{\circ}) sin (4 5^{\circ})

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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{3}{2}

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}

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - \frac{3 2}{2}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= - \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

= - \frac{3}{2}

tan (\frac{π}{6} + \frac{5 π}{4})

cosh (ln (6))

cos^{2} (18 0^{\circ}) - sin^{2} (18 0^{\circ})

tan (7 3^{\circ})

arctan (\frac{12}{16})

What is the value of cos(105)-cos(15) ?
The value of cos(105)-cos(15) is -sqrt(3/2)