What is the value of tan(arccos(24/25)-arcsin(12/13)) ?

The value of tan(arccos(24/25)-arcsin(12/13)) is -253/204

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tan(arccos(24/25)-arcsin(12/13))

Solution

tan (arccos (\frac{24}{25}) - arcsin (\frac{12}{13}))

Solution

- \frac{253}{204}

Decimal Notation

- 1.24019 \dots

Solution steps

tan (arccos (\frac{24}{25}) - arcsin (\frac{12}{13}))

Rewrite using trig identities:

\frac{tan ( arccos ( \frac{24}{25} ) ) - tan ( arcsin ( \frac{12}{13} ) )}{1 + tan ( arccos ( \frac{24}{25} ) ) tan ( arcsin ( \frac{12}{13} ) )}

tan (arccos (\frac{24}{25}) - arcsin (\frac{12}{13}))

Use the Angle Difference identity:

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

= \frac{tan ( arccos ( \frac{24}{25} ) ) - tan ( arcsin ( \frac{12}{13} ) )}{1 + tan ( arccos ( \frac{24}{25} ) ) tan ( arcsin ( \frac{12}{13} ) )}

= \frac{tan ( arccos ( \frac{24}{25} ) ) - tan ( arcsin ( \frac{12}{13} ) )}{1 + tan ( arccos ( \frac{24}{25} ) ) tan ( arcsin ( \frac{12}{13} ) )}

Rewrite using trig identities:

tan (arccos (\frac{24}{25})) = \frac{7}{24}

tan (arccos (\frac{24}{25}))

Rewrite using trig identities:

tan (arccos (\frac{24}{25})) = \frac{1 - ( \frac{24}{25} ) ^{2}}{( \frac{24}{25} )}

Use the following identity:

tan (arccos (x)) = \frac{1 - x ^{2}}{x}

= \frac{1 - ( \frac{24}{25} ) ^{2}}{( \frac{24}{25} )}

= \frac{1 - ( \frac{24}{25} ) ^{2}}{\frac{24}{25}}

Simplify

= \frac{7}{24}

Rewrite using trig identities:

tan (arcsin (\frac{12}{13})) = \frac{12}{5}

tan (arcsin (\frac{12}{13}))

Rewrite using trig identities:

tan (arcsin (\frac{12}{13})) = \frac{( \frac{12}{13} ) 1 - ( \frac{12}{13} ) ^{2}}{1 - ( \frac{12}{13} ) ^{2}}

Use the following identity:

tan (arcsin (x)) = \frac{x 1 - x ^{2}}{1 - x ^{2}}

= \frac{( \frac{12}{13} ) 1 - ( \frac{12}{13} ) ^{2}}{1 - ( \frac{12}{13} ) ^{2}}

= \frac{\frac{12}{13} 1 - ( \frac{12}{13} ) ^{2}}{1 - ( \frac{12}{13} ) ^{2}}

Simplify

= \frac{12}{5}

= \frac{\frac{7}{24} - \frac{12}{5}}{1 + \frac{7}{24} \cdot \frac{12}{5}}

Simplify

\frac{\frac{7}{24} - \frac{12}{5}}{1 + \frac{7}{24} \cdot \frac{12}{5}} : - \frac{253}{204}

\frac{\frac{7}{24} - \frac{12}{5}}{1 + \frac{7}{24} \cdot \frac{12}{5}}

\frac{7}{24} \cdot \frac{12}{5} = \frac{7}{10}

\frac{7}{24} \cdot \frac{12}{5}

Cross-cancel common factor:

12

12, 24

Greatest Common Divisor (GCD)

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Prime factorization of

24 : 2 \cdot 2 \cdot 2 \cdot 3

24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3

The prime factors common to

12, 24

are

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

= \frac{7}{2} \cdot \frac{1}{5}

Multiply fractions:

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

= \frac{7 \cdot 1}{2 \cdot 5}

Multiply the numbers:

7 \cdot 1 = 7

= \frac{7}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{7}{10}

= \frac{\frac{7}{24} - \frac{12}{5}}{1 + \frac{7}{10}}

Join

\frac{7}{24} - \frac{12}{5} : - \frac{253}{120}

\frac{7}{24} - \frac{12}{5}

Least Common Multiplier of

24, 5 : 120

24, 5

Least Common Multiplier (LCM)

Prime factorization of

24 : 2 \cdot 2 \cdot 2 \cdot 3

24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

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

24

5

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 = 120

= 120

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

120

For

\frac{7}{24} :

multiply the denominator and numerator by

5

\frac{7}{24} = \frac{7 \cdot 5}{24 \cdot 5} = \frac{35}{120}

For

\frac{12}{5} :

multiply the denominator and numerator by

24

\frac{12}{5} = \frac{12 \cdot 24}{5 \cdot 24} = \frac{288}{120}

= \frac{35}{120} - \frac{288}{120}

Since the denominators are equal, combine the fractions:

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

= \frac{35 - 288}{120}

Subtract the numbers:

35 - 288 = - 253

= \frac{- 253}{120}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{253}{120}

= \frac{- \frac{253}{120}}{1 + \frac{7}{10}}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{253}{120}}{1 + \frac{7}{10}}

Apply the fraction rule:

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

\frac{\frac{253}{120}}{1 + \frac{7}{10}} = \frac{253}{120 ( 1 + \frac{7}{10} )}

= - \frac{253}{120 ( 1 + \frac{7}{10} )}

Join

1 + \frac{7}{10} : \frac{17}{10}

1 + \frac{7}{10}

Convert element to fraction:

1 = \frac{1 \cdot 10}{10}

= \frac{1 \cdot 10}{10} + \frac{7}{10}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 10 + 7}{10}

1 \cdot 10 + 7 = 17

1 \cdot 10 + 7

Multiply the numbers:

1 \cdot 10 = 10

= 10 + 7

Add the numbers:

10 + 7 = 17

= 17

= \frac{17}{10}

= - \frac{253}{120 \cdot \frac{17}{10}}

Multiply

120 \cdot \frac{17}{10} : 204

120 \cdot \frac{17}{10}

Multiply fractions:

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

= \frac{17 \cdot 120}{10}

Multiply the numbers:

17 \cdot 120 = 2040

= \frac{2040}{10}

Divide the numbers:

\frac{2040}{10} = 204

= 204

= - \frac{253}{204}

= - \frac{253}{204}

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Frequently Asked Questions (FAQ)

What is the value of tan(arccos(24/25)-arcsin(12/13)) ?
The value of tan(arccos(24/25)-arcsin(12/13)) is -253/204