What is the value of cos(arctan(5/7)+arctan(1/6)) ?

The value of cos(arctan(5/7)+arctan(1/6)) is (sqrt(2))/2

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cos(arctan(5/7)+arctan(1/6))

Solution

cos (arctan (\frac{5}{7}) + arctan (\frac{1}{6}))

Solution

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

cos (arctan (\frac{5}{7}) + arctan (\frac{1}{6}))

Rewrite using trig identities:

arctan (\frac{5}{7}) + arctan (\frac{1}{6}) = arctan (1)

arctan (\frac{5}{7}) + arctan (\frac{1}{6})

Use the Sum to Product identity:

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

= arctan (\frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{7} \cdot \frac{1}{6}})

Simplify:

\frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{7} \cdot \frac{1}{6}} = 1

\frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{7} \cdot \frac{1}{6}}

\frac{5}{7} \cdot \frac{1}{6} = \frac{5}{42}

\frac{5}{7} \cdot \frac{1}{6}

Multiply fractions:

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

= \frac{5 \cdot 1}{7 \cdot 6}

Multiply the numbers:

5 \cdot 1 = 5

= \frac{5}{7 \cdot 6}

Multiply the numbers:

7 \cdot 6 = 42

= \frac{5}{42}

= \frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{42}}

Join

\frac{5}{7} + \frac{1}{6} : \frac{37}{42}

\frac{5}{7} + \frac{1}{6}

Least Common Multiplier of

7, 6 : 42

7, 6

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

7

6

= 7 \cdot 2 \cdot 3

Multiply the numbers:

7 \cdot 2 \cdot 3 = 42

= 42

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

42

For

\frac{5}{7} :

multiply the denominator and numerator by

6

\frac{5}{7} = \frac{5 \cdot 6}{7 \cdot 6} = \frac{30}{42}

For

\frac{1}{6} :

multiply the denominator and numerator by

7

\frac{1}{6} = \frac{1 \cdot 7}{6 \cdot 7} = \frac{7}{42}

= \frac{30}{42} + \frac{7}{42}

Since the denominators are equal, combine the fractions:

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

= \frac{30 + 7}{42}

Add the numbers:

30 + 7 = 37

= \frac{37}{42}

= \frac{\frac{37}{42}}{1 - \frac{5}{42}}

Apply the fraction rule:

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

= \frac{37}{42 ( 1 - \frac{5}{42} )}

Join

1 - \frac{5}{42} : \frac{37}{42}

1 - \frac{5}{42}

Convert element to fraction:

1 = \frac{1 \cdot 42}{42}

= \frac{1 \cdot 42}{42} - \frac{5}{42}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 42 - 5}{42}

1 \cdot 42 - 5 = 37

1 \cdot 42 - 5

Multiply the numbers:

1 \cdot 42 = 42

= 42 - 5

Subtract the numbers:

42 - 5 = 37

= 37

= \frac{37}{42}

= \frac{37}{42 \cdot \frac{37}{42}}

Multiply

42 \cdot \frac{37}{42} : 37

42 \cdot \frac{37}{42}

Multiply fractions:

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

= \frac{37 \cdot 42}{42}

Cancel the common factor:

42

= 37

= \frac{37}{37}

Apply rule

\frac{a}{a} = 1

= 1

= arctan (1)

= cos (arctan (1))

Rewrite using trig identities:

cos (arctan (1)) = \frac{1 + 1 ^{2}}{1 + 1 ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{2}{2}

Popular Examples

arctan(1.75)

arctan (1.75)

sqrt((1-cos(150))/2)

\frac{1 - cos ( 15 0 ^{\circ} )}{2}

arctan(7/9)

arctan (\frac{7}{9})

tan(630)

tan (63 0^{\circ})

arctan((-4)/4)

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

Frequently Asked Questions (FAQ)

What is the value of cos(arctan(5/7)+arctan(1/6)) ?
The value of cos(arctan(5/7)+arctan(1/6)) is (sqrt(2))/2