What is the value of cos(240)-cos(0) ?

The value of cos(240)-cos(0) is -3/2

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cos(240)-cos(0)

cos (24 0^{\circ}) - cos (0^{\circ})

- \frac{3}{2}

Decimal Notation

- 1.5

Solution steps

cos (24 0^{\circ}) - cos (0)

Rewrite using trig identities:

- 2 sin^{2} (12 0^{\circ})

cos (24 0^{\circ}) - cos (0)

Use the Sum to Product identity:

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

= - 2 sin (\frac{24 0 ^{\circ} + 0}{2}) sin (\frac{24 0 ^{\circ} - 0}{2})

Simplify

= - 2 sin^{2} (12 0^{\circ})

= - 2 sin^{2} (12 0^{\circ})

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 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}

= - 2 (\frac{3}{2})^{2}

Simplify

- 2 (\frac{3}{2})^{2} : - \frac{3}{2}

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

(\frac{3}{2})^{2} = \frac{3}{2 ^{2}}

(\frac{3}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\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

= 3

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

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

Multiply fractions:

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

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

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{6}{2 ^{2}}

Cancel

\frac{6}{2 ^{2}} : \frac{3}{2}

\frac{6}{2 ^{2}}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel the common factor:

2

= \frac{3}{2}

= - \frac{3}{2}

= - \frac{3}{2}

- cot (- \frac{π}{4})

- sin (3 π)

sec (3 2^{\circ})

arccsc (- 7)

arctan (\frac{4}{- 2})