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

The value of cos(arctan(sqrt(3))-arcsin((sqrt(3))/2)) is 1

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cos(arctan(sqrt(3))-arcsin((sqrt(3))/2))

Solution

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

Solution

1

Solution steps

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

Rewrite using trig identities:

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

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

Use the Angle Difference identity:

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

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

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

Rewrite using trig identities:

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

cos (arctan (3))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

Rewrite using trig identities:

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

sin (arctan (3))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{3}{2}

= \frac{1}{2} \cdot \frac{1}{2} + \frac{3}{2} \cdot \frac{3}{2}

Simplify

\frac{1}{2} \cdot \frac{1}{2} + \frac{3}{2} \cdot \frac{3}{2} : 1

\frac{1}{2} \cdot \frac{1}{2} + \frac{3}{2} \cdot \frac{3}{2}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

\frac{3}{2} \cdot \frac{3}{2} = \frac{3}{4}

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

= \frac{1}{4} + \frac{3}{4}

Apply rule

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

= \frac{1 + 3}{4}

Add the numbers:

1 + 3 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

= 1

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Frequently Asked Questions (FAQ)

What is the value of cos(arctan(sqrt(3))-arcsin((sqrt(3))/2)) ?
The value of cos(arctan(sqrt(3))-arcsin((sqrt(3))/2)) is 1