What is the value of sin(arcsin(5/13)-arccos(-3/5)) ?

The value of sin(arcsin(5/13)-arccos(-3/5)) is -63/65

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sin(arcsin(5/13)-arccos(-3/5))

Solution

sin (arcsin (\frac{5}{13}) - arccos (- \frac{3}{5}))

Solution

- \frac{63}{65}

Decimal Notation

- 0.96923 \dots

Solution steps

sin (arcsin (\frac{5}{13}) - arccos (- \frac{3}{5}))

Rewrite using trig identities:

sin (arcsin (\frac{5}{13})) cos (arccos (- \frac{3}{5})) - cos (arcsin (\frac{5}{13})) sin (arccos (- \frac{3}{5}))

sin (arcsin (\frac{5}{13}) - arccos (- \frac{3}{5}))

Use the Angle Difference identity:

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

= sin (arcsin (\frac{5}{13})) cos (arccos (- \frac{3}{5})) - cos (arcsin (\frac{5}{13})) sin (arccos (- \frac{3}{5}))

= sin (arcsin (\frac{5}{13})) cos (arccos (- \frac{3}{5})) - cos (arcsin (\frac{5}{13})) sin (arccos (- \frac{3}{5}))

Rewrite using trig identities:

sin (arcsin (\frac{5}{13})) = \frac{5}{13}

Use the following identity:

sin (arcsin (x)) = x

= \frac{5}{13}

Rewrite using trig identities:

cos (arccos (- \frac{3}{5})) = - \frac{3}{5}

Use the following identity:

cos (arccos (x)) = x

= - \frac{3}{5}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{12}{13}

Rewrite using trig identities:

sin (arccos (- \frac{3}{5})) = \frac{4}{5}

sin (arccos (- \frac{3}{5}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{4}{5}

= \frac{5}{13} (- \frac{3}{5}) - \frac{12}{13} \cdot \frac{4}{5}

Simplify

\frac{5}{13} (- \frac{3}{5}) - \frac{12}{13} \cdot \frac{4}{5} : - \frac{63}{65}

\frac{5}{13} (- \frac{3}{5}) - \frac{12}{13} \cdot \frac{4}{5}

Remove parentheses:

(- a) = - a

= - \frac{5}{13} \cdot \frac{3}{5} - \frac{12}{13} \cdot \frac{4}{5}

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

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

Cross-cancel:

5

= \frac{3}{13}

\frac{12}{13} \cdot \frac{4}{5} = \frac{48}{65}

\frac{12}{13} \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{12 \cdot 4}{13 \cdot 5}

Multiply the numbers:

12 \cdot 4 = 48

= \frac{48}{13 \cdot 5}

Multiply the numbers:

13 \cdot 5 = 65

= \frac{48}{65}

= - \frac{3}{13} - \frac{48}{65}

Least Common Multiplier of

13, 65 : 65

13, 65

Least Common Multiplier (LCM)

Prime factorization of

13 : 13

13

13

is a prime number, therefore no factorization is possible

= 13

Prime factorization of

65 : 5 \cdot 13

65

65

divides by

565 = 13 \cdot 5

= 5 \cdot 13

5, 13

are all prime numbers, therefore no further factorization is possible

= 5 \cdot 13

Multiply each factor the greatest number of times it occurs in either

13

65

= 13 \cdot 5

Multiply the numbers:

13 \cdot 5 = 65

= 65

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

65

For

\frac{3}{13} :

multiply the denominator and numerator by

5

\frac{3}{13} = \frac{3 \cdot 5}{13 \cdot 5} = \frac{15}{65}

= - \frac{15}{65} - \frac{48}{65}

Since the denominators are equal, combine the fractions:

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

= \frac{- 15 - 48}{65}

Subtract the numbers:

- 15 - 48 = - 63

= \frac{- 63}{65}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{63}{65}

= - \frac{63}{65}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(5/13)-arccos(-3/5)) ?
The value of sin(arcsin(5/13)-arccos(-3/5)) is -63/65