What is the value of arctan(4)-arctan(3/5) ?

The value of arctan(4)-arctan(3/5) is pi/4

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arctan(4)-arctan(3/5)

Solution

arctan (4) - arctan (\frac{3}{5})

Solution

\frac{π}{4}

Decimal Notation

45

Solution steps

arctan (4) - arctan (\frac{3}{5})

Rewrite using trig identities:

arctan (1)

arctan (4) - arctan (\frac{3}{5})

Use the Sum to Product identity:

arctan (s) - arctan (t) = arctan (\frac{s - t}{1 + s t})

= arctan (\frac{4 - \frac{3}{5}}{1 + 4 \cdot \frac{3}{5}})

Simplify:

\frac{4 - \frac{3}{5}}{1 + 4 \cdot \frac{3}{5}} = 1

\frac{4 - \frac{3}{5}}{1 + 4 \cdot \frac{3}{5}}

4 \cdot \frac{3}{5} = \frac{12}{5}

4 \cdot \frac{3}{5}

Multiply fractions:

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

= \frac{3 \cdot 4}{5}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12}{5}

= \frac{4 - \frac{3}{5}}{1 + \frac{12}{5}}

Join

4 - \frac{3}{5} : \frac{17}{5}

4 - \frac{3}{5}

Convert element to fraction:

4 = \frac{4 \cdot 5}{5}

= \frac{4 \cdot 5}{5} - \frac{3}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{4 \cdot 5 - 3}{5}

4 \cdot 5 - 3 = 17

4 \cdot 5 - 3

Multiply the numbers:

4 \cdot 5 = 20

= 20 - 3

Subtract the numbers:

20 - 3 = 17

= 17

= \frac{17}{5}

= \frac{\frac{17}{5}}{1 + \frac{12}{5}}

Apply the fraction rule:

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

= \frac{17}{5 ( 1 + \frac{12}{5} )}

Join

1 + \frac{12}{5} : \frac{17}{5}

1 + \frac{12}{5}

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= \frac{1 \cdot 5}{5} + \frac{12}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 5 + 12}{5}

1 \cdot 5 + 12 = 17

1 \cdot 5 + 12

Multiply the numbers:

1 \cdot 5 = 5

= 5 + 12

Add the numbers:

5 + 12 = 17

= 17

= \frac{17}{5}

= \frac{17}{5 \cdot \frac{17}{5}}

Multiply

5 \cdot \frac{17}{5} : 17

5 \cdot \frac{17}{5}

Multiply fractions:

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

= \frac{17 \cdot 5}{5}

Cancel the common factor:

5

= 17

= \frac{17}{17}

Apply rule

\frac{a}{a} = 1

= 1

= arctan (1)

= arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

arctan (1)

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4}

= \frac{π}{4}

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Frequently Asked Questions (FAQ)

What is the value of arctan(4)-arctan(3/5) ?
The value of arctan(4)-arctan(3/5) is pi/4