What is the value of tan(arcsin(5/13)+pi/6) ?

The value of tan(arcsin(5/13)+pi/6) is (240+169sqrt(3))/(407)

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Popular Trigonometry >

tan(arcsin(5/13)+pi/6)

Solution

tan (arcsin (\frac{5}{13}) + \frac{π}{6})

Solution

\frac{240 + 169 3}{407}

Decimal

1.30888 \dots

Solution steps

tan (arcsin (\frac{5}{13}) + \frac{π}{6})

Rewrite using trig identities:

\frac{tan ( arcsin ( \frac{5}{13} ) ) + tan ( \frac{π}{6} )}{1 - tan ( arcsin ( \frac{5}{13} ) ) tan ( \frac{π}{6} )}

tan (arcsin (\frac{5}{13}) + \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{5}{13} ) ) + tan ( \frac{π}{6} )}{1 - tan ( arcsin ( \frac{5}{13} ) ) tan ( \frac{π}{6} )}

= \frac{tan ( arcsin ( \frac{5}{13} ) ) + tan ( \frac{π}{6} )}{1 - tan ( arcsin ( \frac{5}{13} ) ) tan ( \frac{π}{6} )}

Rewrite using trig identities:

tan (arcsin (\frac{5}{13})) = \frac{5}{12}

tan (arcsin (\frac{5}{13}))

Rewrite using trig identities:

tan (arcsin (\frac{5}{13})) = \frac{( \frac{5}{13} ) 1 - ( \frac{5}{13} ) ^{2}}{1 - ( \frac{5}{13} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{5}{13} ) 1 - ( \frac{5}{13} ) ^{2}}{1 - ( \frac{5}{13} ) ^{2}}

= \frac{\frac{5}{13} 1 - ( \frac{5}{13} ) ^{2}}{1 - ( \frac{5}{13} ) ^{2}}

Simplify

= \frac{5}{12}

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{5}{12} + \frac{3}{3}}{1 - \frac{5}{12} \cdot \frac{3}{3}}

Simplify

\frac{\frac{5}{12} + \frac{3}{3}}{1 - \frac{5}{12} \cdot \frac{3}{3}} : \frac{240 + 169 3}{407}

\frac{\frac{5}{12} + \frac{3}{3}}{1 - \frac{5}{12} \cdot \frac{3}{3}}

\frac{5}{12} \cdot \frac{3}{3} = \frac{5 3}{36}

\frac{5}{12} \cdot \frac{3}{3}

Multiply fractions:

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

= \frac{5 3}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{5 3}{36}

= \frac{\frac{5}{12} + \frac{3}{3}}{1 - \frac{5 3}{36}}

Join

\frac{5}{12} + \frac{3}{3} : \frac{5 + 4 3}{12}

\frac{5}{12} + \frac{3}{3}

Least Common Multiplier of

12, 3 : 12

12, 3

Least Common Multiplier (LCM)

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

12

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}{3} :

multiply the denominator and numerator by

4

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

= \frac{5}{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{5 + 3 \cdot 4}{12}

= \frac{\frac{5 + 4 3}{12}}{1 - \frac{5 3}{36}}

Apply the fraction rule:

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

= \frac{5 + 3 \cdot 4}{12 ( 1 - \frac{5 3}{36} )}

Join

1 - \frac{5 3}{36} : \frac{36 - 5 3}{36}

1 - \frac{5 3}{36}

Convert element to fraction:

1 = \frac{1 \cdot 36}{36}

= \frac{1 \cdot 36}{36} - \frac{5 3}{36}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 36 - 5 3}{36}

Multiply the numbers:

1 \cdot 36 = 36

= \frac{36 - 5 3}{36}

= \frac{5 + 4 3}{12 \cdot \frac{36 - 5 3}{36}}

Multiply

12 \cdot \frac{36 - 5 3}{36} : \frac{36 - 5 3}{3}

12 \cdot \frac{36 - 5 3}{36}

Multiply fractions:

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

= \frac{( 36 - 5 3 ) \cdot 12}{36}

Cancel the common factor:

12

= \frac{36 - 5 3}{3}

= \frac{5 + 4 3}{\frac{36 - 5 3}{3}}

Apply the fraction rule:

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

= \frac{( 5 + 3 \cdot 4 ) \cdot 3}{36 - 5 3}

Rationalize

\frac{3 ( 5 + 4 3 )}{36 - 5 3} : \frac{240 + 169 3}{407}

\frac{3 ( 5 + 4 3 )}{36 - 5 3}

Multiply by the conjugate

\frac{36 + 5 3}{36 + 5 3}

= \frac{( 5 + 3 \cdot 4 ) \cdot 3 ( 36 + 5 3 )}{( 36 - 5 3 ) ( 36 + 5 3 )}

(5 + 3 \cdot 4) \cdot 3 (36 + 53) = 720 + 5073

(5 + 3 \cdot 4) \cdot 3 (36 + 53)

= 3 (5 + 43) (36 + 53)

Expand

(5 + 3 \cdot 4) (36 + 53) : 240 + 1693

(5 + 3 \cdot 4) (36 + 53)

Apply FOIL method:

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

a = 5, b = 3 \cdot 4, c = 36, d = 53

= 5 \cdot 36 + 5 \cdot 53 + 3 \cdot 4 \cdot 36 + 3 \cdot 4 \cdot 53

= 5 \cdot 36 + 5 \cdot 53 + 4 \cdot 363 + 4 \cdot 533

Simplify

5 \cdot 36 + 5 \cdot 53 + 4 \cdot 363 + 4 \cdot 533 : 240 + 1693

5 \cdot 36 + 5 \cdot 53 + 4 \cdot 363 + 4 \cdot 533

5 \cdot 36 = 180

5 \cdot 36

Multiply the numbers:

5 \cdot 36 = 180

= 180

5 \cdot 53 = 253

5 \cdot 53

Multiply the numbers:

5 \cdot 5 = 25

= 253

4 \cdot 363 = 1443

4 \cdot 363

Multiply the numbers:

4 \cdot 36 = 144

= 1443

4 \cdot 533 = 60

4 \cdot 533

Multiply the numbers:

4 \cdot 5 = 20

= 2033

Apply radical rule:

a a = a

33 = 3

= 20 \cdot 3

Multiply the numbers:

20 \cdot 3 = 60

= 60

= 180 + 253 + 1443 + 60

Add similar elements:

253 + 1443 = 1693

= 180 + 1693 + 60

Add the numbers:

180 + 60 = 240

= 240 + 1693

= 240 + 1693

= 3 (240 + 1693)

Expand

3 (240 + 1693) : 720 + 5073

3 (240 + 1693)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 240, c = 1693

= 3 \cdot 240 + 3 \cdot 1693

Simplify

3 \cdot 240 + 3 \cdot 1693 : 720 + 5073

3 \cdot 240 + 3 \cdot 1693

Multiply the numbers:

3 \cdot 240 = 720

= 720 + 3 \cdot 1693

Multiply the numbers:

3 \cdot 169 = 507

= 720 + 5073

= 720 + 5073

= 720 + 5073

(36 - 53) (36 + 53) = 1221

(36 - 53) (36 + 53)

Apply Difference of Two Squares Formula:

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

a = 36, b = 53

= 3 6^{2} - (53)^{2}

Simplify

3 6^{2} - (53)^{2} : 1221

3 6^{2} - (53)^{2}

3 6^{2} = 1296

3 6^{2}

3 6^{2} = 1296

= 1296

(53)^{2} = 75

(53)^{2}

Apply exponent rule:

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

= 5^{2} (3)^{2}

(3)^{2} : 3

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

= 5^{2} \cdot 3

5^{2} = 25

= 25 \cdot 3

Multiply the numbers:

25 \cdot 3 = 75

= 75

= 1296 - 75

Subtract the numbers:

1296 - 75 = 1221

= 1221

= 1221

= \frac{720 + 507 3}{1221}

Factor

720 + 5073 : 3 (240 + 1693)

720 + 5073

Rewrite as

= 3 \cdot 240 + 3 \cdot 1693

Factor out common term

3

= 3 (240 + 1693)

= \frac{3 ( 240 + 169 3 )}{1221}

Cancel the common factor:

3

= \frac{240 + 169 3}{407}

= \frac{240 + 169 3}{407}

= \frac{240 + 169 3}{407}

Popular Examples

cos(2arccos(-(sqrt(2))/2))

Frequently Asked Questions (FAQ)

What is the value of tan(arcsin(5/13)+pi/6) ?
The value of tan(arcsin(5/13)+pi/6) is (240+169sqrt(3))/(407)