What is the value of tan(arcsin(2/3)-arccos(3/5)) ?

The value of tan(arcsin(2/3)-arccos(3/5)) is -(2(25sqrt(5)-54))/(19)

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tan(arcsin(2/3)-arccos(3/5))

Solution

tan (arcsin (\frac{2}{3}) - arccos (\frac{3}{5}))

Solution

- \frac{2 ( 25 5 - 54 )}{19}

Decimal Notation

- 0.20017 \dots

Solution steps

tan (arcsin (\frac{2}{3}) - arccos (\frac{3}{5}))

Rewrite using trig identities:

\frac{tan ( arcsin ( \frac{2}{3} ) ) - tan ( arccos ( \frac{3}{5} ) )}{1 + tan ( arcsin ( \frac{2}{3} ) ) tan ( arccos ( \frac{3}{5} ) )}

tan (arcsin (\frac{2}{3}) - arccos (\frac{3}{5}))

Use the Angle Difference identity:

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

= \frac{tan ( arcsin ( \frac{2}{3} ) ) - tan ( arccos ( \frac{3}{5} ) )}{1 + tan ( arcsin ( \frac{2}{3} ) ) tan ( arccos ( \frac{3}{5} ) )}

= \frac{tan ( arcsin ( \frac{2}{3} ) ) - tan ( arccos ( \frac{3}{5} ) )}{1 + tan ( arcsin ( \frac{2}{3} ) ) tan ( arccos ( \frac{3}{5} ) )}

Rewrite using trig identities:

tan (arcsin (\frac{2}{3})) = \frac{2 5}{5}

tan (arcsin (\frac{2}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 5}{5}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

tan (arccos (\frac{3}{5})) = \frac{1 - ( \frac{3}{5} ) ^{2}}{( \frac{3}{5} )}

Use the following identity:

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

= \frac{1 - ( \frac{3}{5} ) ^{2}}{( \frac{3}{5} )}

= \frac{1 - ( \frac{3}{5} ) ^{2}}{\frac{3}{5}}

Simplify

= \frac{4}{3}

= \frac{\frac{2 5}{5} - \frac{4}{3}}{1 + \frac{2 5}{5} \cdot \frac{4}{3}}

Simplify

\frac{\frac{2 5}{5} - \frac{4}{3}}{1 + \frac{2 5}{5} \cdot \frac{4}{3}} : - \frac{2 ( 25 5 - 54 )}{19}

\frac{\frac{2 5}{5} - \frac{4}{3}}{1 + \frac{2 5}{5} \cdot \frac{4}{3}}

\frac{2 5}{5} \cdot \frac{4}{3} = \frac{8 5}{15}

\frac{2 5}{5} \cdot \frac{4}{3}

Multiply fractions:

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

= \frac{2 5 \cdot 4}{5 \cdot 3}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8 5}{5 \cdot 3}

Multiply the numbers:

5 \cdot 3 = 15

= \frac{8 5}{15}

= \frac{\frac{2 5}{5} - \frac{4}{3}}{1 + \frac{8 5}{15}}

Join

\frac{2 5}{5} - \frac{4}{3} : \frac{6 5 - 20}{15}

\frac{2 5}{5} - \frac{4}{3}

Least Common Multiplier of

5, 3 : 15

5, 3

Least Common Multiplier (LCM)

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

5

3

= 5 \cdot 3

Multiply the numbers:

5 \cdot 3 = 15

= 15

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

15

For

\frac{2 5}{5} :

multiply the denominator and numerator by

3

\frac{2 5}{5} = \frac{2 5 \cdot 3}{5 \cdot 3} = \frac{6 5}{15}

For

\frac{4}{3} :

multiply the denominator and numerator by

5

\frac{4}{3} = \frac{4 \cdot 5}{3 \cdot 5} = \frac{20}{15}

= \frac{6 5}{15} - \frac{20}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{6 5 - 20}{15}

= \frac{\frac{6 5 - 20}{15}}{1 + \frac{8 5}{15}}

Apply the fraction rule:

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

= \frac{6 5 - 20}{15 ( 1 + \frac{8 5}{15} )}

Join

1 + \frac{8 5}{15} : \frac{15 + 8 5}{15}

1 + \frac{8 5}{15}

Convert element to fraction:

1 = \frac{1 \cdot 15}{15}

= \frac{1 \cdot 15}{15} + \frac{8 5}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 15 + 8 5}{15}

Multiply the numbers:

1 \cdot 15 = 15

= \frac{15 + 8 5}{15}

= \frac{6 5 - 20}{15 \cdot \frac{15 + 8 5}{15}}

Multiply

15 \cdot \frac{15 + 8 5}{15} : 15 + 85

15 \cdot \frac{15 + 8 5}{15}

Multiply fractions:

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

= \frac{( 15 + 8 5 ) \cdot 15}{15}

Cancel the common factor:

15

= 15 + 85

= \frac{6 5 - 20}{15 + 8 5}

Rationalize

\frac{6 5 - 20}{15 + 8 5} : - \frac{2 ( 25 5 - 54 )}{19}

\frac{6 5 - 20}{15 + 8 5}

Multiply by the conjugate

\frac{15 - 8 5}{15 - 8 5}

= \frac{( 6 5 - 20 ) ( 15 - 8 5 )}{( 15 + 8 5 ) ( 15 - 8 5 )}

(65 - 20) (15 - 85) = 2505 - 540

(65 - 20) (15 - 85)

Apply FOIL method:

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

a = 65, b = - 20, c = 15, d = - 85

= 65 \cdot 15 + 65 (- 85) + (- 20) \cdot 15 + (- 20) (- 85)

Apply minus-plus rules

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

= 6 \cdot 155 - 6 \cdot 855 - 20 \cdot 15 + 20 \cdot 85

Simplify

6 \cdot 155 - 6 \cdot 855 - 20 \cdot 15 + 20 \cdot 85 : 2505 - 540

6 \cdot 155 - 6 \cdot 855 - 20 \cdot 15 + 20 \cdot 85

6 \cdot 155 = 905

6 \cdot 155

Multiply the numbers:

6 \cdot 15 = 90

= 905

6 \cdot 855 = 240

6 \cdot 855

Multiply the numbers:

6 \cdot 8 = 48

= 4855

Apply radical rule:

a a = a

55 = 5

= 48 \cdot 5

Multiply the numbers:

48 \cdot 5 = 240

= 240

20 \cdot 15 = 300

20 \cdot 15

Multiply the numbers:

20 \cdot 15 = 300

= 300

20 \cdot 85 = 1605

20 \cdot 85

Multiply the numbers:

20 \cdot 8 = 160

= 1605

= 905 - 240 - 300 + 1605

Add similar elements:

905 + 1605 = 2505

= 2505 - 240 - 300

Subtract the numbers:

- 240 - 300 = - 540

= 2505 - 540

= 2505 - 540

(15 + 85) (15 - 85) = - 95

(15 + 85) (15 - 85)

Apply Difference of Two Squares Formula:

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

a = 15, b = 85

= 1 5^{2} - (85)^{2}

Simplify

1 5^{2} - (85)^{2} : - 95

1 5^{2} - (85)^{2}

1 5^{2} = 225

1 5^{2}

1 5^{2} = 225

= 225

(85)^{2} = 320

(85)^{2}

Apply exponent rule:

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

= 8^{2} (5)^{2}

(5)^{2} : 5

Apply radical rule:

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

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

Apply exponent rule:

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

= 5^{\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

= 5

= 8^{2} \cdot 5

8^{2} = 64

= 64 \cdot 5

Multiply the numbers:

64 \cdot 5 = 320

= 320

= 225 - 320

Subtract the numbers:

225 - 320 = - 95

= - 95

= - 95

= \frac{250 5 - 540}{- 95}

Apply the fraction rule:

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

= - \frac{250 5 - 540}{95}

Cancel

\frac{250 5 - 540}{95} : \frac{2 ( 25 5 - 54 )}{19}

\frac{250 5 - 540}{95}

Factor

2505 - 540 : 10 (255 - 54)

2505 - 540

Rewrite as

= 10 \cdot 255 - 10 \cdot 54

Factor out common term

10

= 10 (255 - 54)

= \frac{10 ( 25 5 - 54 )}{95}

Cancel the common factor:

5

= \frac{2 ( 25 5 - 54 )}{19}

= - \frac{2 ( 25 5 - 54 )}{19}

= - \frac{2 ( 25 5 - 54 )}{19}

= - \frac{2 ( 25 5 - 54 )}{19}

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Frequently Asked Questions (FAQ)

What is the value of tan(arcsin(2/3)-arccos(3/5)) ?
The value of tan(arcsin(2/3)-arccos(3/5)) is -(2(25sqrt(5)-54))/(19)