What is the value of tan(arcsin(3/4)-arccos(1/5)) ?

The value of tan(arcsin(3/4)-arccos(1/5)) is -(75sqrt(7)-32sqrt(6))/(209)

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tan(arcsin(3/4)-arccos(1/5))

Solution

tan (arcsin (\frac{3}{4}) - arccos (\frac{1}{5}))

Solution

- \frac{75 7 - 32 6}{209}

Decimal Notation

- 0.57439 \dots

Solution steps

tan (arcsin (\frac{3}{4}) - arccos (\frac{1}{5}))

Rewrite using trig identities:

\frac{tan ( arcsin ( \frac{3}{4} ) ) - tan ( arccos ( \frac{1}{5} ) )}{1 + tan ( arcsin ( \frac{3}{4} ) ) tan ( arccos ( \frac{1}{5} ) )}

tan (arcsin (\frac{3}{4}) - arccos (\frac{1}{5}))

Use the Angle Difference identity:

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

= \frac{tan ( arcsin ( \frac{3}{4} ) ) - tan ( arccos ( \frac{1}{5} ) )}{1 + tan ( arcsin ( \frac{3}{4} ) ) tan ( arccos ( \frac{1}{5} ) )}

= \frac{tan ( arcsin ( \frac{3}{4} ) ) - tan ( arccos ( \frac{1}{5} ) )}{1 + tan ( arcsin ( \frac{3}{4} ) ) tan ( arccos ( \frac{1}{5} ) )}

Rewrite using trig identities:

tan (arcsin (\frac{3}{4})) = \frac{3 7}{7}

tan (arcsin (\frac{3}{4}))

Rewrite using trig identities:

tan (arcsin (\frac{3}{4})) = \frac{( \frac{3}{4} ) 1 - ( \frac{3}{4} ) ^{2}}{1 - ( \frac{3}{4} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{3}{4} ) 1 - ( \frac{3}{4} ) ^{2}}{1 - ( \frac{3}{4} ) ^{2}}

= \frac{\frac{3}{4} 1 - ( \frac{3}{4} ) ^{2}}{1 - ( \frac{3}{4} ) ^{2}}

Simplify

= \frac{3 7}{7}

Rewrite using trig identities:

tan (arccos (\frac{1}{5})) = 26

tan (arccos (\frac{1}{5}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= 26

= \frac{\frac{3 7}{7} - 2 6}{1 + \frac{3 7}{7} \cdot 2 6}

Simplify

\frac{\frac{3 7}{7} - 2 6}{1 + \frac{3 7}{7} \cdot 2 6} : - \frac{75 7 - 32 6}{209}

\frac{\frac{3 7}{7} - 2 6}{1 + \frac{3 7}{7} \cdot 2 6}

\frac{3 7}{7} \cdot 26 = \frac{6 42}{7}

\frac{3 7}{7} \cdot 26

Multiply fractions:

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

= \frac{3 7 \cdot 2 6}{7}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 7 6}{7}

Simplify

676 : 642

676

Apply radical rule:

a b = a \cdot b

76 = 7 \cdot 6

= 6 7 \cdot 6

Multiply the numbers:

7 \cdot 6 = 42

= 642

= \frac{6 42}{7}

= \frac{\frac{3 7}{7} - 2 6}{1 + \frac{6 42}{7}}

Join

\frac{3 7}{7} - 26 : \frac{3 7 - 14 6}{7}

\frac{3 7}{7} - 26

Convert element to fraction:

26 = \frac{2 \cdot 6 \cdot 7}{7}

= \frac{3 7}{7} - \frac{2 6 \cdot 7}{7}

Since the denominators are equal, combine the fractions:

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

= \frac{3 7 - 2 6 \cdot 7}{7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{3 7 - 14 6}{7}

= \frac{\frac{3 7 - 14 6}{7}}{1 + \frac{6 42}{7}}

Apply the fraction rule:

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

= \frac{3 7 - 14 6}{7 ( 1 + \frac{6 42}{7} )}

Join

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

1 + \frac{6 42}{7}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7 + 6 42}{7}

= \frac{3 7 - 14 6}{7 \cdot \frac{7 + 6 42}{7}}

Multiply

7 \cdot \frac{7 + 6 42}{7} : 7 + 642

7 \cdot \frac{7 + 6 42}{7}

Multiply fractions:

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

= \frac{( 7 + 6 42 ) \cdot 7}{7}

Cancel the common factor:

7

= 7 + 642

= \frac{3 7 - 14 6}{7 + 6 42}

Rationalize

\frac{3 7 - 14 6}{7 + 6 42} : - \frac{75 7 - 32 6}{209}

\frac{3 7 - 14 6}{7 + 6 42}

Multiply by the conjugate

\frac{7 - 6 42}{7 - 6 42}

= \frac{( 3 7 - 14 6 ) ( 7 - 6 42 )}{( 7 + 6 42 ) ( 7 - 6 42 )}

(37 - 146) (7 - 642) = 5257 - 2246

(37 - 146) (7 - 642)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 37, b = - 146, c = 7, d = - 642

= 37 \cdot 7 + 37 (- 642) + (- 146) \cdot 7 + (- 146) (- 642)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 3 \cdot 77 - 3 \cdot 6742 - 14 \cdot 76 + 14 \cdot 6642

Simplify

3 \cdot 77 - 3 \cdot 6742 - 14 \cdot 76 + 14 \cdot 6642 : 5257 - 2246

3 \cdot 77 - 3 \cdot 6742 - 14 \cdot 76 + 14 \cdot 6642

3 \cdot 77 = 217

3 \cdot 77

Multiply the numbers:

3 \cdot 7 = 21

= 217

3 \cdot 6742 = 1266

3 \cdot 6742

Factor integer

42 = 7 \cdot 6

= 3 \cdot 67 7 \cdot 6

Apply radical rule:

7 \cdot 6 = 76

= 3 \cdot 6776

Apply radical rule:

a a = a

77 = 7

= 3 \cdot 6 \cdot 76

Multiply the numbers:

3 \cdot 6 \cdot 7 = 126

= 1266

14 \cdot 76 = 986

14 \cdot 76

Multiply the numbers:

14 \cdot 7 = 98

= 986

14 \cdot 6642 = 5047

14 \cdot 6642

Factor integer

14 = 2 \cdot 7

= 2 \cdot 7 \cdot 6642

Factor integer

6 = 2 \cdot 3

= 2 \cdot 7 \cdot 2 \cdot 3642

Factor integer

6 = 2 \cdot 3

= 2 \cdot 7 \cdot 2 \cdot 3 2 \cdot 3 42

Apply radical rule:

2 \cdot 3 = 23

= 2 \cdot 7 \cdot 2 \cdot 32342

Factor integer

42 = 2 \cdot 3 \cdot 7

= 2 \cdot 7 \cdot 2 \cdot 323 2 \cdot 3 \cdot 7

Apply radical rule:

2 \cdot 3 \cdot 7 = 237

= 2 \cdot 7 \cdot 2 \cdot 323237

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= 7 \cdot 2^{1 + 1} \cdot 323237

Add the numbers:

1 + 1 = 2

= 7 \cdot 2^{2} \cdot 323237

Apply radical rule:

a a = a

22 = 2

= 2^{2} \cdot 7 \cdot 3 \cdot 2337

Apply radical rule:

a a = a

33 = 3

= 2^{2} \cdot 7 \cdot 3 \cdot 2 \cdot 37

Multiply the numbers:

7 \cdot 3 \cdot 2 \cdot 3 = 126

= 2^{2} \cdot 1267

2^{2} = 4

= 126 \cdot 47

Multiply the numbers:

126 \cdot 4 = 504

= 5047

= 217 - 1266 - 986 + 5047

Add similar elements:

- 1266 - 986 = - 2246

= 217 - 2246 + 5047

Add similar elements:

217 + 5047 = 5257

= 5257 - 2246

= 5257 - 2246

(7 + 642) (7 - 642) = - 1463

(7 + 642) (7 - 642)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 7, b = 642

= 7^{2} - (642)^{2}

Simplify

7^{2} - (642)^{2} : - 1463

7^{2} - (642)^{2}

7^{2} = 49

7^{2}

7^{2} = 49

= 49

(642)^{2} = 1512

(642)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} (42)^{2}

(42)^{2} : 42

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 4 2^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 42

= 6^{2} \cdot 42

6^{2} = 36

= 36 \cdot 42

Multiply the numbers:

36 \cdot 42 = 1512

= 1512

= 49 - 1512

Subtract the numbers:

49 - 1512 = - 1463

= - 1463

= - 1463

= \frac{525 7 - 224 6}{- 1463}

Apply the fraction rule:

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

= - \frac{525 7 - 224 6}{1463}

Cancel

\frac{525 7 - 224 6}{1463} : \frac{75 7 - 32 6}{209}

\frac{525 7 - 224 6}{1463}

Factor

5257 - 2246 : 7 (757 - 326)

5257 - 2246

Rewrite as

= 7 \cdot 757 - 7 \cdot 326

Factor out common term

7

= 7 (757 - 326)

= \frac{7 ( 75 7 - 32 6 )}{1463}

Cancel the common factor:

7

= \frac{75 7 - 32 6}{209}

= - \frac{75 7 - 32 6}{209}

= - \frac{75 7 - 32 6}{209}

= - \frac{75 7 - 32 6}{209}

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Frequently Asked Questions (FAQ)

What is the value of tan(arcsin(3/4)-arccos(1/5)) ?
The value of tan(arcsin(3/4)-arccos(1/5)) is -(75sqrt(7)-32sqrt(6))/(209)