What is the value of tan(arcsin(3/5)+pi/6) ?

The value of tan(arcsin(3/5)+pi/6) is (48+25sqrt(3))/(39)

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Popular Trigonometry >

tan(arcsin(3/5)+pi/6)

Solution

tan (arcsin (\frac{3}{5}) + \frac{π}{6})

Solution

\frac{48 + 25 3}{39}

Decimal

2.34105 \dots

Solution steps

tan (arcsin (\frac{3}{5}) + \frac{π}{6})

Rewrite using trig identities:

\frac{tan ( arcsin ( \frac{3}{5} ) ) + tan ( \frac{π}{6} )}{1 - tan ( arcsin ( \frac{3}{5} ) ) tan ( \frac{π}{6} )}

tan (arcsin (\frac{3}{5}) + \frac{π}{6})

Use the Angle Sum identity:

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

= \frac{tan ( arcsin ( \frac{3}{5} ) ) + tan ( \frac{π}{6} )}{1 - tan ( arcsin ( \frac{3}{5} ) ) tan ( \frac{π}{6} )}

= \frac{tan ( arcsin ( \frac{3}{5} ) ) + tan ( \frac{π}{6} )}{1 - tan ( arcsin ( \frac{3}{5} ) ) tan ( \frac{π}{6} )}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{4}

Use the following trivial identity:

tan (\frac{π}{6}) = \frac{3}{3}

tan (\frac{π}{6})

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

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

Simplify

\frac{\frac{3}{4} + \frac{3}{3}}{1 - \frac{3}{4} \cdot \frac{3}{3}} : \frac{48 + 25 3}{39}

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

Multiply

\frac{3}{4} \cdot \frac{3}{3} : \frac{3}{4}

\frac{3}{4} \cdot \frac{3}{3}

Cross-cancel:

3

= \frac{3}{4}

= \frac{\frac{3}{4} + \frac{3}{3}}{1 - \frac{3}{4}}

Join

\frac{3}{4} + \frac{3}{3} : \frac{9 + 4 3}{12}

\frac{3}{4} + \frac{3}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{3}{4} :

multiply the denominator and numerator by

3

\frac{3}{4} = \frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12}

For

\frac{3}{3} :

multiply the denominator and numerator by

4

\frac{3}{3} = \frac{3 \cdot 4}{3 \cdot 4} = \frac{3 \cdot 4}{12}

= \frac{9}{12} + \frac{3 \cdot 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{9 + 3 \cdot 4}{12}

= \frac{\frac{9 + 4 3}{12}}{1 - \frac{3}{4}}

Apply the fraction rule:

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

= \frac{9 + 3 \cdot 4}{12 ( 1 - \frac{3}{4} )}

Join

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

1 - \frac{3}{4}

Convert element to fraction:

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

= \frac{1 \cdot 4}{4} - \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 3}{4}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 - 3}{4}

= \frac{9 + 4 3}{12 \cdot \frac{4 - 3}{4}}

Multiply

12 \cdot \frac{4 - 3}{4} : 3 (4 - 3)

12 \cdot \frac{4 - 3}{4}

Multiply fractions:

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

= \frac{( 4 - 3 ) \cdot 12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3 (4 - 3)

= \frac{9 + 4 3}{3 ( 4 - 3 )}

Rationalize

\frac{9 + 4 3}{3 ( 4 - 3 )} : \frac{48 + 25 3}{39}

\frac{9 + 4 3}{3 ( 4 - 3 )}

Multiply by the conjugate

\frac{4 + 3}{4 + 3}

= \frac{( 9 + 3 \cdot 4 ) ( 4 + 3 )}{3 ( 4 - 3 ) ( 4 + 3 )}

(9 + 3 \cdot 4) (4 + 3) = 48 + 253

(9 + 3 \cdot 4) (4 + 3)

= (9 + 43) (4 + 3)

Apply FOIL method:

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

a = 9, b = 3 \cdot 4, c = 4, d = 3

= 9 \cdot 4 + 93 + 3 \cdot 4 \cdot 4 + 3 \cdot 43

= 9 \cdot 4 + 93 + 4 \cdot 43 + 433

Simplify

9 \cdot 4 + 93 + 4 \cdot 43 + 433 : 48 + 253

9 \cdot 4 + 93 + 4 \cdot 43 + 433

9 \cdot 4 = 36

9 \cdot 4

Multiply the numbers:

9 \cdot 4 = 36

= 36

4 \cdot 43 = 163

4 \cdot 43

Multiply the numbers:

4 \cdot 4 = 16

= 163

433 = 12

433

Apply radical rule:

a a = a

33 = 3

= 4 \cdot 3

Multiply the numbers:

4 \cdot 3 = 12

= 12

= 36 + 93 + 163 + 12

Add similar elements:

93 + 163 = 253

= 36 + 253 + 12

Add the numbers:

36 + 12 = 48

= 48 + 253

= 48 + 253

3 (4 - 3) (4 + 3) = 39

3 (4 - 3) (4 + 3)

Expand

(4 - 3) (4 + 3) : 13

(4 - 3) (4 + 3)

Apply Difference of Two Squares Formula:

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

a = 4, b = 3

= 4^{2} - (3)^{2}

Simplify

4^{2} - (3)^{2} : 13

4^{2} - (3)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 3^{\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

= 3

= 16 - 3

Subtract the numbers:

16 - 3 = 13

= 13

= 13

= 3 \cdot 13

Expand

3 \cdot 13 : 39

3 \cdot 13

Distribute parentheses

= 3 \cdot 13

Multiply the numbers:

3 \cdot 13 = 39

= 39

= 39

= \frac{48 + 25 3}{39}

= \frac{48 + 25 3}{39}

= \frac{48 + 25 3}{39}

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Frequently Asked Questions (FAQ)

What is the value of tan(arcsin(3/5)+pi/6) ?
The value of tan(arcsin(3/5)+pi/6) is (48+25sqrt(3))/(39)