What is the value of sin(arcsin(1/2)+arccos(5/13)) ?

The value of sin(arcsin(1/2)+arccos(5/13)) is 5/26+(6sqrt(3))/(13)

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sin(arcsin(1/2)+arccos(5/13))

Solution

sin (arcsin (\frac{1}{2}) + arccos (\frac{5}{13}))

Solution

\frac{5}{26} + \frac{6 3}{13}

Decimal Notation

0.99171 \dots

Solution steps

sin (arcsin (\frac{1}{2}) + arccos (\frac{5}{13}))

Rewrite using trig identities:

sin (arcsin (\frac{1}{2})) cos (arccos (\frac{5}{13})) + cos (arcsin (\frac{1}{2})) sin (arccos (\frac{5}{13}))

sin (arcsin (\frac{1}{2}) + arccos (\frac{5}{13}))

Use the Angle Sum identity:

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

= sin (arcsin (\frac{1}{2})) cos (arccos (\frac{5}{13})) + cos (arcsin (\frac{1}{2})) sin (arccos (\frac{5}{13}))

= sin (arcsin (\frac{1}{2})) cos (arccos (\frac{5}{13})) + cos (arcsin (\frac{1}{2})) sin (arccos (\frac{5}{13}))

Rewrite using trig identities:

sin (arcsin (\frac{1}{2})) = \frac{1}{2}

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{2}

Rewrite using trig identities:

cos (arccos (\frac{5}{13})) = \frac{5}{13}

Use the following identity:

cos (arccos (x)) = x

= \frac{5}{13}

Rewrite using trig identities:

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

cos (arcsin (\frac{1}{2}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{12}{13}

= \frac{1}{2} \cdot \frac{5}{13} + \frac{3}{2} \cdot \frac{12}{13}

Simplify

\frac{1}{2} \cdot \frac{5}{13} + \frac{3}{2} \cdot \frac{12}{13} : \frac{5}{26} + \frac{6 3}{13}

\frac{1}{2} \cdot \frac{5}{13} + \frac{3}{2} \cdot \frac{12}{13}

\frac{1}{2} \cdot \frac{5}{13} = \frac{5}{26}

\frac{1}{2} \cdot \frac{5}{13}

Multiply fractions:

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

= \frac{1 \cdot 5}{2 \cdot 13}

Multiply the numbers:

1 \cdot 5 = 5

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

Multiply the numbers:

2 \cdot 13 = 26

= \frac{5}{26}

\frac{3}{2} \cdot \frac{12}{13} = \frac{6 3}{13}

\frac{3}{2} \cdot \frac{12}{13}

Cross-cancel common factor:

2

= \frac{3}{1} \cdot \frac{6}{13}

Multiply fractions:

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

= \frac{3 \cdot 6}{1 \cdot 13}

Multiply the numbers:

1 \cdot 13 = 13

= \frac{6 3}{13}

= \frac{5}{26} + \frac{6 3}{13}

= \frac{5}{26} + \frac{6 3}{13}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(1/2)+arccos(5/13)) ?
The value of sin(arcsin(1/2)+arccos(5/13)) is 5/26+(6sqrt(3))/(13)