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

The value of tan(arcsin(3/4)-arccos(1/3)) is -(27sqrt(7)-32sqrt(2))/(65)

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

Solution

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

Solution

- \frac{27 7 - 32 2}{65}

Decimal Notation

- 0.40277 \dots

Solution steps

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

Rewrite using trig identities:

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

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

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

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

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}{3})) = 22

tan (arccos (\frac{1}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= 22

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

Simplify

\frac{\frac{3 7}{7} - 2 2}{1 + \frac{3 7}{7} \cdot 2 2} : - \frac{27 7 - 32 2}{65}

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

\frac{3 7}{7} \cdot 22 = \frac{6 14}{7}

\frac{3 7}{7} \cdot 22

Multiply fractions:

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

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

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 7 2}{7}

Simplify

672 : 614

672

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 6 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 614

= \frac{6 14}{7}

= \frac{\frac{3 7}{7} - 2 2}{1 + \frac{6 14}{7}}

Join

\frac{3 7}{7} - 22 : \frac{3 7 - 14 2}{7}

\frac{3 7}{7} - 22

Convert element to fraction:

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

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

Multiply the numbers:

2 \cdot 7 = 14

= \frac{3 7 - 14 2}{7}

= \frac{\frac{3 7 - 14 2}{7}}{1 + \frac{6 14}{7}}

Apply the fraction rule:

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

= \frac{3 7 - 14 2}{7 ( 1 + \frac{6 14}{7} )}

Join

1 + \frac{6 14}{7} : \frac{7 + 6 14}{7}

1 + \frac{6 14}{7}

Convert element to fraction:

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

= \frac{1 \cdot 7}{7} + \frac{6 14}{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 14}{7}

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7 + 6 14}{7}

= \frac{3 7 - 14 2}{7 \cdot \frac{7 + 6 14}{7}}

Multiply

7 \cdot \frac{7 + 6 14}{7} : 7 + 614

7 \cdot \frac{7 + 6 14}{7}

Multiply fractions:

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

= \frac{( 7 + 6 14 ) \cdot 7}{7}

Cancel the common factor:

7

= 7 + 614

= \frac{3 7 - 14 2}{7 + 6 14}

Rationalize

\frac{3 7 - 14 2}{7 + 6 14} : - \frac{27 7 - 32 2}{65}

\frac{3 7 - 14 2}{7 + 6 14}

Multiply by the conjugate

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

= \frac{( 3 7 - 14 2 ) ( 7 - 6 14 )}{( 7 + 6 14 ) ( 7 - 6 14 )}

(37 - 142) (7 - 614) = 1897 - 2242

(37 - 142) (7 - 614)

Apply FOIL method:

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

a = 37, b = - 142, c = 7, d = - 614

= 37 \cdot 7 + 37 (- 614) + (- 142) \cdot 7 + (- 142) (- 614)

Apply minus-plus rules

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

= 3 \cdot 77 - 3 \cdot 6714 - 14 \cdot 72 + 14 \cdot 6214

Simplify

3 \cdot 77 - 3 \cdot 6714 - 14 \cdot 72 + 14 \cdot 6214 : 1897 - 2242

3 \cdot 77 - 3 \cdot 6714 - 14 \cdot 72 + 14 \cdot 6214

3 \cdot 77 = 217

3 \cdot 77

Multiply the numbers:

3 \cdot 7 = 21

= 217

3 \cdot 6714 = 1262

3 \cdot 6714

Factor integer

14 = 7 \cdot 2

= 3 \cdot 67 7 \cdot 2

Apply radical rule:

7 \cdot 2 = 72

= 3 \cdot 6772

Apply radical rule:

a a = a

77 = 7

= 3 \cdot 6 \cdot 72

Multiply the numbers:

3 \cdot 6 \cdot 7 = 126

= 1262

14 \cdot 72 = 982

14 \cdot 72

Multiply the numbers:

14 \cdot 7 = 98

= 982

14 \cdot 6214 = 1687

14 \cdot 6214

Factor integer

14 = 2 \cdot 7

= 2 \cdot 7 \cdot 6214

Factor integer

6 = 2 \cdot 3

= 2 \cdot 7 \cdot 2 \cdot 3214

Factor integer

14 = 2 \cdot 7

= 2 \cdot 7 \cdot 2 \cdot 32 2 \cdot 7

Apply radical rule:

2 \cdot 7 = 27

= 2 \cdot 7 \cdot 2 \cdot 3227

Apply exponent rule:

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

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

= 7 \cdot 2^{1 + 1} \cdot 3227

Add the numbers:

1 + 1 = 2

= 7 \cdot 2^{2} \cdot 3227

Apply radical rule:

a a = a

22 = 2

= 2^{2} \cdot 7 \cdot 3 \cdot 27

Multiply the numbers:

7 \cdot 3 \cdot 2 = 42

= 2^{2} \cdot 427

2^{2} = 4

= 42 \cdot 47

Multiply the numbers:

42 \cdot 4 = 168

= 1687

= 217 - 1262 - 982 + 1687

Add similar elements:

- 1262 - 982 = - 2242

= 217 - 2242 + 1687

Add similar elements:

217 + 1687 = 1897

= 1897 - 2242

= 1897 - 2242

(7 + 614) (7 - 614) = - 455

(7 + 614) (7 - 614)

Apply Difference of Two Squares Formula:

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

a = 7, b = 614

= 7^{2} - (614)^{2}

Simplify

7^{2} - (614)^{2} : - 455

7^{2} - (614)^{2}

7^{2} = 49

7^{2}

7^{2} = 49

= 49

(614)^{2} = 504

(614)^{2}

Apply exponent rule:

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

= 6^{2} (14)^{2}

(14)^{2} : 14

Apply radical rule:

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

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

Apply exponent rule:

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

= 1 4^{\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

= 14

= 6^{2} \cdot 14

6^{2} = 36

= 36 \cdot 14

Multiply the numbers:

36 \cdot 14 = 504

= 504

= 49 - 504

Subtract the numbers:

49 - 504 = - 455

= - 455

= - 455

= \frac{189 7 - 224 2}{- 455}

Apply the fraction rule:

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

= - \frac{189 7 - 224 2}{455}

Cancel

\frac{189 7 - 224 2}{455} : \frac{27 7 - 32 2}{65}

\frac{189 7 - 224 2}{455}

Factor

1897 - 2242 : 7 (277 - 322)

1897 - 2242

Rewrite as

= 7 \cdot 277 - 7 \cdot 322

Factor out common term

7

= 7 (277 - 322)

= \frac{7 ( 27 7 - 32 2 )}{455}

Cancel the common factor:

7

= \frac{27 7 - 32 2}{65}

= - \frac{27 7 - 32 2}{65}

= - \frac{27 7 - 32 2}{65}

= - \frac{27 7 - 32 2}{65}

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Frequently Asked Questions (FAQ)

What is the value of tan(arcsin(3/4)-arccos(1/3)) ?
The value of tan(arcsin(3/4)-arccos(1/3)) is -(27sqrt(7)-32sqrt(2))/(65)