What is the value of cos(arcsin(2/3)+arctan(3/2)) ?

The value of cos(arcsin(2/3)+arctan(3/2)) is (2sqrt(65))/(39)-(2sqrt(13))/(13)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(arcsin(2/3)+arctan(3/2))

Solution

cos (arcsin (\frac{2}{3}) + arctan (\frac{3}{2}))

Solution

\frac{2 65}{39} - \frac{2 13}{13}

Decimal Notation

- 0.14125 \dots

Solution steps

cos (arcsin (\frac{2}{3}) + arctan (\frac{3}{2}))

Rewrite using trig identities:

cos (arcsin (\frac{2}{3})) cos (arctan (\frac{3}{2})) - sin (arcsin (\frac{2}{3})) sin (arctan (\frac{3}{2}))

cos (arcsin (\frac{2}{3}) + arctan (\frac{3}{2}))

Use the Angle Sum identity:

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

= cos (arcsin (\frac{2}{3})) cos (arctan (\frac{3}{2})) - sin (arcsin (\frac{2}{3})) sin (arctan (\frac{3}{2}))

= cos (arcsin (\frac{2}{3})) cos (arctan (\frac{3}{2})) - sin (arcsin (\frac{2}{3})) sin (arctan (\frac{3}{2}))

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{5}{3}

Rewrite using trig identities:

cos (arctan (\frac{3}{2})) = \frac{2 13}{13}

cos (arctan (\frac{3}{2}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 13}{13}

Rewrite using trig identities:

sin (arcsin (\frac{2}{3})) = \frac{2}{3}

Use the following identity:

sin (arcsin (x)) = x

= \frac{2}{3}

Rewrite using trig identities:

sin (arctan (\frac{3}{2})) = \frac{3 13}{13}

sin (arctan (\frac{3}{2}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3 13}{13}

= \frac{5}{3} \cdot \frac{2 13}{13} - \frac{2}{3} \cdot \frac{3 13}{13}

Simplify

\frac{5}{3} \cdot \frac{2 13}{13} - \frac{2}{3} \cdot \frac{3 13}{13} : \frac{2 65}{39} - \frac{2 13}{13}

\frac{5}{3} \cdot \frac{2 13}{13} - \frac{2}{3} \cdot \frac{3 13}{13}

\frac{5}{3} \cdot \frac{2 13}{13} = \frac{2 65}{39}

\frac{5}{3} \cdot \frac{2 13}{13}

Multiply fractions:

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

= \frac{5 \cdot 2 13}{3 \cdot 13}

Multiply the numbers:

3 \cdot 13 = 39

= \frac{2 5 13}{39}

Simplify

5 \cdot 213 : 265

5 \cdot 213

Apply radical rule:

a b = a \cdot b

513 = 5 \cdot 13

= 2 5 \cdot 13

Multiply the numbers:

5 \cdot 13 = 65

= 265

= \frac{2 65}{39}

\frac{2}{3} \cdot \frac{3 13}{13} = \frac{2 13}{13}

\frac{2}{3} \cdot \frac{3 13}{13}

Multiply fractions:

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

= \frac{2 \cdot 3 13}{3 \cdot 13}

Cancel the common factor:

3

= \frac{2 13}{13}

= \frac{2 65}{39} - \frac{2 13}{13}

= \frac{2 65}{39} - \frac{2 13}{13}

Popular Examples

2sin(55)

2 sin (5 5^{\circ})

sin(325)

sin (32 5^{\circ})

arccos(0.534)

arccos (0.534)

sinh(1)-sinh(-1)

sinh (1) - sinh (- 1)

sin(pi)1

sin (π) 1

Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(2/3)+arctan(3/2)) ?
The value of cos(arcsin(2/3)+arctan(3/2)) is (2sqrt(65))/(39)-(2sqrt(13))/(13)