What is the general solution for arctan(x+2)=arcsin(7/25)+arccos(4/5) ?

The general solution for arctan(x+2)=arcsin(7/25)+arccos(4/5) is x=-2/3

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Popular Trigonometry >

arctan(x+2)=arcsin(7/25)+arccos(4/5)

Solution

arctan (x + 2) = arcsin (\frac{7}{25}) + arccos (\frac{4}{5})

Solution

x = - \frac{2}{3}

Solution steps

arctan (x + 2) = arcsin (\frac{7}{25}) + arccos (\frac{4}{5})

Apply trig inverse properties

arctan (x + 2) = arcsin (\frac{7}{25}) + arccos (\frac{4}{5})

arctan (x) = a \Rightarrow x = tan (a)

x + 2 = tan (arcsin (\frac{7}{25}) + arccos (\frac{4}{5}))

tan (arcsin (\frac{7}{25}) + arccos (\frac{4}{5})) = \frac{4}{3}

tan (arcsin (\frac{7}{25}) + arccos (\frac{4}{5}))

Rewrite using trig identities:

\frac{tan ( arcsin ( \frac{7}{25} ) ) + tan ( arccos ( \frac{4}{5} ) )}{1 - tan ( arcsin ( \frac{7}{25} ) ) tan ( arccos ( \frac{4}{5} ) )}

tan (arcsin (\frac{7}{25}) + arccos (\frac{4}{5}))

Use the Angle Sum identity:

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

= \frac{tan ( arcsin ( \frac{7}{25} ) ) + tan ( arccos ( \frac{4}{5} ) )}{1 - tan ( arcsin ( \frac{7}{25} ) ) tan ( arccos ( \frac{4}{5} ) )}

= \frac{tan ( arcsin ( \frac{7}{25} ) ) + tan ( arccos ( \frac{4}{5} ) )}{1 - tan ( arcsin ( \frac{7}{25} ) ) tan ( arccos ( \frac{4}{5} ) )}

Rewrite using trig identities:

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

tan (arcsin (\frac{7}{25}))

Rewrite using trig identities:

tan (arcsin (\frac{7}{25})) = \frac{( \frac{7}{25} ) 1 - ( \frac{7}{25} ) ^{2}}{1 - ( \frac{7}{25} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{7}{25} ) 1 - ( \frac{7}{25} ) ^{2}}{1 - ( \frac{7}{25} ) ^{2}}

= \frac{\frac{7}{25} 1 - ( \frac{7}{25} ) ^{2}}{1 - ( \frac{7}{25} ) ^{2}}

Simplify

= \frac{7}{24}

Rewrite using trig identities:

tan (arccos (\frac{4}{5})) = \frac{3}{4}

tan (arccos (\frac{4}{5}))

Rewrite using trig identities:

tan (arccos (\frac{4}{5})) = \frac{1 - ( \frac{4}{5} ) ^{2}}{( \frac{4}{5} )}

Use the following identity:

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

= \frac{1 - ( \frac{4}{5} ) ^{2}}{( \frac{4}{5} )}

= \frac{1 - ( \frac{4}{5} ) ^{2}}{\frac{4}{5}}

Simplify

= \frac{3}{4}

= \frac{\frac{7}{24} + \frac{3}{4}}{1 - \frac{7}{24} \cdot \frac{3}{4}}

Simplify

\frac{\frac{7}{24} + \frac{3}{4}}{1 - \frac{7}{24} \cdot \frac{3}{4}} : \frac{4}{3}

\frac{\frac{7}{24} + \frac{3}{4}}{1 - \frac{7}{24} \cdot \frac{3}{4}}

\frac{7}{24} \cdot \frac{3}{4} = \frac{7}{32}

\frac{7}{24} \cdot \frac{3}{4}

Cross-cancel common factor:

3

3, 24

Greatest Common Divisor (GCD)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 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

3, 24

are

= 3

= \frac{7}{8} \cdot \frac{1}{4}

Multiply fractions:

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

= \frac{7 \cdot 1}{8 \cdot 4}

Multiply the numbers:

7 \cdot 1 = 7

= \frac{7}{8 \cdot 4}

Multiply the numbers:

8 \cdot 4 = 32

= \frac{7}{32}

= \frac{\frac{7}{24} + \frac{3}{4}}{1 - \frac{7}{32}}

Join

\frac{7}{24} + \frac{3}{4} : \frac{25}{24}

\frac{7}{24} + \frac{3}{4}

Least Common Multiplier of

24, 4 : 24

24, 4

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

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

24

4

= 2 \cdot 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 3 = 24

= 24

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

24

For

\frac{3}{4} :

multiply the denominator and numerator by

6

\frac{3}{4} = \frac{3 \cdot 6}{4 \cdot 6} = \frac{18}{24}

= \frac{7}{24} + \frac{18}{24}

Since the denominators are equal, combine the fractions:

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

= \frac{7 + 18}{24}

Add the numbers:

7 + 18 = 25

= \frac{25}{24}

= \frac{\frac{25}{24}}{1 - \frac{7}{32}}

Apply the fraction rule:

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

= \frac{25}{24 ( 1 - \frac{7}{32} )}

Join

1 - \frac{7}{32} : \frac{25}{32}

1 - \frac{7}{32}

Convert element to fraction:

1 = \frac{1 \cdot 32}{32}

= \frac{1 \cdot 32}{32} - \frac{7}{32}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 32 - 7}{32}

1 \cdot 32 - 7 = 25

1 \cdot 32 - 7

Multiply the numbers:

1 \cdot 32 = 32

= 32 - 7

Subtract the numbers:

32 - 7 = 25

= 25

= \frac{25}{32}

= \frac{25}{24 \cdot \frac{25}{32}}

Multiply

24 \cdot \frac{25}{32} : \frac{75}{4}

24 \cdot \frac{25}{32}

Multiply fractions:

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

= \frac{25 \cdot 24}{32}

Multiply the numbers:

25 \cdot 24 = 600

= \frac{600}{32}

Cancel the common factor:

8

= \frac{75}{4}

= \frac{25}{\frac{75}{4}}

Apply the fraction rule:

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

= \frac{25 \cdot 4}{75}

Multiply the numbers:

25 \cdot 4 = 100

= \frac{100}{75}

Cancel the common factor:

25

= \frac{4}{3}

= \frac{4}{3}

x + 2 = \frac{4}{3}

x + 2 = \frac{4}{3}

Solve

x + 2 = \frac{4}{3} : x = - \frac{2}{3}

x + 2 = \frac{4}{3}

Move

2

to the right side

x + 2 = \frac{4}{3}

Subtract

2

from both sides

x + 2 - 2 = \frac{4}{3} - 2

Simplify

x + 2 - 2 = \frac{4}{3} - 2

Simplify

x + 2 - 2 : x

x + 2 - 2

Add similar elements:

2 - 2 = 0

= x

Simplify

\frac{4}{3} - 2 : - \frac{2}{3}

\frac{4}{3} - 2

Convert element to fraction:

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

= - \frac{2 \cdot 3}{3} + \frac{4}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 \cdot 3 + 4}{3}

- 2 \cdot 3 + 4 = - 2

- 2 \cdot 3 + 4

Multiply the numbers:

2 \cdot 3 = 6

= - 6 + 4

Add/Subtract the numbers:

- 6 + 4 = - 2

= - 2

= \frac{- 2}{3}

Apply the fraction rule:

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

= - \frac{2}{3}

x = - \frac{2}{3}

x = - \frac{2}{3}

x = - \frac{2}{3}

x = - \frac{2}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for arctan(x+2)=arcsin(7/25)+arccos(4/5) ?
The general solution for arctan(x+2)=arcsin(7/25)+arccos(4/5) is x=-2/3