What is the value of cos(arcsin(5/6)+arctan(4/3)) ?

The value of cos(arcsin(5/6)+arctan(4/3)) is (sqrt(11))/(10)-2/3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(arcsin(5/6)+arctan(4/3))

Solution

cos (arcsin (\frac{5}{6}) + arctan (\frac{4}{3}))

Solution

\frac{11}{10} - \frac{2}{3}

Decimal Notation

- 0.33500 \dots

Solution steps

cos (arcsin (\frac{5}{6}) + arctan (\frac{4}{3}))

Rewrite using trig identities:

cos (arcsin (\frac{5}{6})) cos (arctan (\frac{4}{3})) - sin (arcsin (\frac{5}{6})) sin (arctan (\frac{4}{3}))

cos (arcsin (\frac{5}{6}) + arctan (\frac{4}{3}))

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (arcsin (\frac{5}{6})) cos (arctan (\frac{4}{3})) - sin (arcsin (\frac{5}{6})) sin (arctan (\frac{4}{3}))

= cos (arcsin (\frac{5}{6})) cos (arctan (\frac{4}{3})) - sin (arcsin (\frac{5}{6})) sin (arctan (\frac{4}{3}))

Rewrite using trig identities:

cos (arcsin (\frac{5}{6})) = \frac{11}{6}

cos (arcsin (\frac{5}{6}))

Rewrite using trig identities:

cos (arcsin (\frac{5}{6})) = 1 - (\frac{5}{6})^{2}

Use the following identity:

cos (arcsin (x)) = 1 - x^{2}

= 1 - (\frac{5}{6})^{2}

= 1 - (\frac{5}{6})^{2}

Simplify

= \frac{11}{6}

Rewrite using trig identities:

cos (arctan (\frac{4}{3})) = \frac{3}{5}

cos (arctan (\frac{4}{3}))

Rewrite using trig identities:

cos (arctan (\frac{4}{3})) = \frac{1 + ( \frac{4}{3} ) ^{2}}{1 + ( \frac{4}{3} ) ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{3}{5}

Rewrite using trig identities:

sin (arcsin (\frac{5}{6})) = \frac{5}{6}

Use the following identity:

sin (arcsin (x)) = x

= \frac{5}{6}

Rewrite using trig identities:

sin (arctan (\frac{4}{3})) = \frac{4}{5}

sin (arctan (\frac{4}{3}))

Rewrite using trig identities:

sin (arctan (\frac{4}{3})) = \frac{( \frac{4}{3} ) 1 + ( \frac{4}{3} ) ^{2}}{1 + ( \frac{4}{3} ) ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{4}{5}

= \frac{11}{6} \cdot \frac{3}{5} - \frac{5}{6} \cdot \frac{4}{5}

Simplify

\frac{11}{6} \cdot \frac{3}{5} - \frac{5}{6} \cdot \frac{4}{5} : \frac{11}{10} - \frac{2}{3}

\frac{11}{6} \cdot \frac{3}{5} - \frac{5}{6} \cdot \frac{4}{5}

\frac{11}{6} \cdot \frac{3}{5} = \frac{11}{10}

\frac{11}{6} \cdot \frac{3}{5}

Cross-cancel common factor:

3

3, 6

Greatest Common Divisor (GCD)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

The prime factors common to

3, 6

are

= 3

= \frac{11}{2} \cdot \frac{1}{5}

Multiply fractions:

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

= \frac{11 \cdot 1}{2 \cdot 5}

Multiply:

11 \cdot 1 = 11

= \frac{11}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{11}{10}

\frac{5}{6} \cdot \frac{4}{5} = \frac{2}{3}

\frac{5}{6} \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{5 \cdot 4}{6 \cdot 5}

Cancel the common factor:

5

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

= \frac{11}{10} - \frac{2}{3}

= \frac{11}{10} - \frac{2}{3}

Popular Examples

csc(27)

csc (2 7^{\circ})

sin^2(16)

sin^{2} (1 6^{\circ})

sec(245)

sec (24 5^{\circ})

cos(135)+isin(135)

cos (13 5^{\circ}) + i sin (13 5^{\circ})

arccot(0.9)

arccot (0.9)

Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(5/6)+arctan(4/3)) ?
The value of cos(arcsin(5/6)+arctan(4/3)) is (sqrt(11))/(10)-2/3