What is the value of cos((5pi)/4)-cos(pi/4) ?

The value of cos((5pi)/4)-cos(pi/4) is -sqrt(2)

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Popular Trigonometry >

cos((5pi)/4)-cos(pi/4)

Solution

cos (\frac{5 π}{4}) - cos (\frac{π}{4})

Solution

- 2

Decimal

- 1.41421 \dots

Solution steps

cos (\frac{5 π}{4}) - cos (\frac{π}{4})

Rewrite using trig identities:

- 2 sin (\frac{3 π}{4}) sin (\frac{π}{2})

cos (\frac{5 π}{4}) - cos (\frac{π}{4})

Use the Sum to Product identity:

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

= - 2 sin (\frac{\frac{5 π}{4} + \frac{π}{4}}{2}) sin (\frac{\frac{5 π}{4} - \frac{π}{4}}{2})

Simplify:

\frac{\frac{5 π}{4} + \frac{π}{4}}{2} = \frac{3 π}{4}

\frac{\frac{5 π}{4} + \frac{π}{4}}{2}

Combine the fractions

\frac{5 π}{4} + \frac{π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{5 π + π}{4}

Add similar elements:

5 π + π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

Simplify:

\frac{\frac{5 π}{4} - \frac{π}{4}}{2} = \frac{π}{2}

\frac{\frac{5 π}{4} - \frac{π}{4}}{2}

Combine the fractions

\frac{5 π}{4} - \frac{π}{4} : π

Apply rule

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

= \frac{5 π - π}{4}

Add similar elements:

5 π - π = 4 π

= \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= π

= \frac{π}{2}

= - 2 sin (\frac{3 π}{4}) sin (\frac{π}{2})

= - 2 sin (\frac{3 π}{4}) sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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}

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - 1 \cdot 2

Multiply:

1 \cdot 2 = 2

= - 2

= - 2

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Frequently Asked Questions (FAQ)

What is the value of cos((5pi)/4)-cos(pi/4) ?
The value of cos((5pi)/4)-cos(pi/4) is -sqrt(2)