What is the general solution for arccos(x)=arcsin(1/4)+arcsin(7/8) ?

The general solution for arccos(x)=arcsin(1/4)+arcsin(7/8) is x= 1/4

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arccos(x)=arcsin(1/4)+arcsin(7/8)

Solution

arccos (x) = arcsin (\frac{1}{4}) + arcsin (\frac{7}{8})

Solution

x = \frac{1}{4}

Solution steps

arccos (x) = arcsin (\frac{1}{4}) + arcsin (\frac{7}{8})

Apply trig inverse properties

arccos (x) = arcsin (\frac{1}{4}) + arcsin (\frac{7}{8})

arccos (x) = a \Rightarrow x = cos (a)

x = cos (arcsin (\frac{1}{4}) + arcsin (\frac{7}{8}))

cos (arcsin (\frac{1}{4}) + arcsin (\frac{7}{8})) = \frac{1}{4}

cos (arcsin (\frac{1}{4}) + arcsin (\frac{7}{8}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{4})) cos (arcsin (\frac{7}{8})) - sin (arcsin (\frac{1}{4})) sin (arcsin (\frac{7}{8}))

cos (arcsin (\frac{1}{4}) + arcsin (\frac{7}{8}))

Use the Angle Sum identity:

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

= cos (arcsin (\frac{1}{4})) cos (arcsin (\frac{7}{8})) - sin (arcsin (\frac{1}{4})) sin (arcsin (\frac{7}{8}))

= cos (arcsin (\frac{1}{4})) cos (arcsin (\frac{7}{8})) - sin (arcsin (\frac{1}{4})) sin (arcsin (\frac{7}{8}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{4})) = \frac{15}{4}

cos (arcsin (\frac{1}{4}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{4})) = 1 - (\frac{1}{4})^{2}

Use the following identity:

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

= 1 - (\frac{1}{4})^{2}

= 1 - (\frac{1}{4})^{2}

Simplify

= \frac{15}{4}

Rewrite using trig identities:

cos (arcsin (\frac{7}{8})) = \frac{15}{8}

cos (arcsin (\frac{7}{8}))

Rewrite using trig identities:

cos (arcsin (\frac{7}{8})) = 1 - (\frac{7}{8})^{2}

Use the following identity:

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

= 1 - (\frac{7}{8})^{2}

= 1 - (\frac{7}{8})^{2}

Simplify

= \frac{15}{8}

Rewrite using trig identities:

sin (arcsin (\frac{1}{4})) = \frac{1}{4}

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{4}

Rewrite using trig identities:

sin (arcsin (\frac{7}{8})) = \frac{7}{8}

Use the following identity:

sin (arcsin (x)) = x

= \frac{7}{8}

= \frac{15}{4} \cdot \frac{15}{8} - \frac{1}{4} \cdot \frac{7}{8}

Simplify

\frac{15}{4} \cdot \frac{15}{8} - \frac{1}{4} \cdot \frac{7}{8} : \frac{1}{4}

\frac{15}{4} \cdot \frac{15}{8} - \frac{1}{4} \cdot \frac{7}{8}

\frac{15}{4} \cdot \frac{15}{8} = \frac{15}{32}

\frac{15}{4} \cdot \frac{15}{8}

Multiply fractions:

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

= \frac{15 15}{4 \cdot 8}

1515 = 15

1515

Apply radical rule:

a a = a

1515 = 15

= 15

= \frac{15}{4 \cdot 8}

Multiply the numbers:

4 \cdot 8 = 32

= \frac{15}{32}

\frac{1}{4} \cdot \frac{7}{8} = \frac{7}{32}

\frac{1}{4} \cdot \frac{7}{8}

Multiply fractions:

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

= \frac{1 \cdot 7}{4 \cdot 8}

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7}{4 \cdot 8}

Multiply the numbers:

4 \cdot 8 = 32

= \frac{7}{32}

= \frac{15}{32} - \frac{7}{32}

Apply rule

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

= \frac{15 - 7}{32}

Subtract the numbers:

15 - 7 = 8

= \frac{8}{32}

Cancel the common factor:

8

= \frac{1}{4}

= \frac{1}{4}

x = \frac{1}{4}

x = \frac{1}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for arccos(x)=arcsin(1/4)+arcsin(7/8) ?
The general solution for arccos(x)=arcsin(1/4)+arcsin(7/8) is x= 1/4