What is the value of sin(arctan(12/5)+arccos(12/13)) ?

The value of sin(arctan(12/5)+arccos(12/13)) is 1

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sin(arctan(12/5)+arccos(12/13))

Solution

sin (arctan (\frac{12}{5}) + arccos (\frac{12}{13}))

Solution

1

Solution steps

sin (arctan (\frac{12}{5}) + arccos (\frac{12}{13}))

Rewrite using trig identities:

sin (arctan (\frac{12}{5})) cos (arccos (\frac{12}{13})) + cos (arctan (\frac{12}{5})) sin (arccos (\frac{12}{13}))

sin (arctan (\frac{12}{5}) + arccos (\frac{12}{13}))

Use the Angle Sum identity:

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

= sin (arctan (\frac{12}{5})) cos (arccos (\frac{12}{13})) + cos (arctan (\frac{12}{5})) sin (arccos (\frac{12}{13}))

= sin (arctan (\frac{12}{5})) cos (arccos (\frac{12}{13})) + cos (arctan (\frac{12}{5})) sin (arccos (\frac{12}{13}))

Rewrite using trig identities:

sin (arctan (\frac{12}{5})) = \frac{12}{13}

sin (arctan (\frac{12}{5}))

Rewrite using trig identities:

sin (arctan (\frac{12}{5})) = \frac{( \frac{12}{5} ) 1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{12}{5} ) 1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

= \frac{\frac{12}{5} 1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Simplify

= \frac{12}{13}

Rewrite using trig identities:

cos (arccos (\frac{12}{13})) = \frac{12}{13}

Use the following identity:

cos (arccos (x)) = x

= \frac{12}{13}

Rewrite using trig identities:

cos (arctan (\frac{12}{5})) = \frac{5}{13}

cos (arctan (\frac{12}{5}))

Rewrite using trig identities:

cos (arctan (\frac{12}{5})) = \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

= \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Simplify

= \frac{5}{13}

Rewrite using trig identities:

sin (arccos (\frac{12}{13})) = \frac{5}{13}

sin (arccos (\frac{12}{13}))

Rewrite using trig identities:

sin (arccos (\frac{12}{13})) = 1 - (\frac{12}{13})^{2}

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

= 1 - (\frac{12}{13})^{2}

= 1 - (\frac{12}{13})^{2}

Simplify

= \frac{5}{13}

= \frac{12}{13} \cdot \frac{12}{13} + \frac{5}{13} \cdot \frac{5}{13}

Simplify

\frac{12}{13} \cdot \frac{12}{13} + \frac{5}{13} \cdot \frac{5}{13} : 1

\frac{12}{13} \cdot \frac{12}{13} + \frac{5}{13} \cdot \frac{5}{13}

\frac{12}{13} \cdot \frac{12}{13} = \frac{144}{169}

\frac{12}{13} \cdot \frac{12}{13}

Multiply fractions:

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

= \frac{12 \cdot 12}{13 \cdot 13}

Multiply the numbers:

12 \cdot 12 = 144

= \frac{144}{13 \cdot 13}

Multiply the numbers:

13 \cdot 13 = 169

= \frac{144}{169}

\frac{5}{13} \cdot \frac{5}{13} = \frac{25}{169}

\frac{5}{13} \cdot \frac{5}{13}

Multiply fractions:

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

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

Multiply the numbers:

5 \cdot 5 = 25

= \frac{25}{13 \cdot 13}

Multiply the numbers:

13 \cdot 13 = 169

= \frac{25}{169}

= \frac{144}{169} + \frac{25}{169}

Apply rule

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

= \frac{144 + 25}{169}

Add the numbers:

144 + 25 = 169

= \frac{169}{169}

Apply rule

\frac{a}{a} = 1

= 1

= 1

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Frequently Asked Questions (FAQ)

What is the value of sin(arctan(12/5)+arccos(12/13)) ?
The value of sin(arctan(12/5)+arccos(12/13)) is 1