The right-angled triangle in the diagram is such that $ ext{sin } x = rac{2}{ ext{√}11}$ and $0 < x < rac{ ext{π}}{4}$. (a) Find the exact value of: (i) $ ext{sin } 2x$ (ii) $ ext{cos } 2x$. (b) By expressing $ ext{sin } 3x$ as $ ext{sin }(2x + x)$, find the exact value of $ ext{sin } 3x$.

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The right-angled triangle in the diagram is such that $ ext{sin } x = rac{2}{ ext{√}11}$ and $0 < x < rac{ ext{π}}{4}$ - Scottish Highers Maths - Question 13 - 2018

Question 13

The right-angled triangle in the diagram is such that $ ext{sin } x = rac{2}{ ext{√}11}$ and $0 < x < rac{ ext{π}}{4}$. (a) Find the exact value of: (i) $ ext{si... show full transcript

Worked Solution & Example Answer:The right-angled triangle in the diagram is such that $ ext{sin } x = rac{2}{ ext{√}11}$ and $0 < x < rac{ ext{π}}{4}$ - Scottish Highers Maths - Question 13 - 2018

Step 1

Find the exact value of: (i) sin 2x

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Answer

To find $ext{sin } 2x$ , we use the double angle formula:

$ext{sin } 2x = 2 imes ext{sin } x imes ext{cos } x$

First, we need to calculate $ext{cos } x$ . Using the Pythagorean identity:

$ext{cos } x = ext{√}(1 - ext{sin}^2 x)$

Substituting $ext{sin } x = rac{2}{ ext{\sqrt}11}$ :

$ext{cos } x = ext{√}igg(1 - igg( rac{2}{ ext{√}11}igg)^2igg) = ext{√}igg(1 - rac{4}{11}igg) = ext{√}igg( rac{7}{11}igg) = rac{ ext{√}7}{ ext{√}11}$

Now substituting the values we have:

$ext{sin } 2x = 2 imes rac{2}{ ext{\sqrt}11} imes rac{ ext{\sqrt}7}{ ext{\sqrt}11} = rac{4 ext{\sqrt}7}{11}$

Step 2

Find the exact value of: (ii) cos 2x

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Answer

To find $ext{cos } 2x$ , we use the double angle formula:

$ext{cos } 2x = ext{cos}^2 x - ext{sin}^2 x$

Substituting the values:

$ext{cos}^2 x = rac{7}{11}, ext{sin}^2 x = rac{4}{11}$

Then,

$ext{cos } 2x = rac{7}{11} - rac{4}{11} = rac{3}{11}$

Step 3

By expressing sin 3x as sin(2x + x), find the exact value of sin 3x

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Answer

Using the angle addition formula for sine:

$ext{sin }(2x + x) = ext{sin } 2x imes ext{cos } x + ext{cos } 2x imes ext{sin } x$

We already calculated:

$ext{sin } 2x = rac{4 ext{\sqrt}7}{11}$
$ext{cos } x = rac{ ext{\sqrt}7}{ ext{\sqrt}11}$
$ext{cos } 2x = rac{3}{11}$
$ext{sin } x = rac{2}{ ext{\sqrt}11}$

Now substituting these values:

$ext{sin } 3x = rac{4 ext{\sqrt}7}{11} imes rac{ ext{\sqrt}7}{ ext{\sqrt}11} + rac{3}{11} imes rac{2}{ ext{\sqrt}11}$

Simplifying:

$ext{sin } 3x = rac{4 imes 7}{11 imes ext{\sqrt}11} + rac{6}{11 imes ext{\sqrt}11} = rac{28 + 6}{11 imes ext{\sqrt}11} = rac{34}{11 imes ext{\sqrt}11}$

The right-angled triangle in the diagram is such that $ ext{sin } x = rac{2}{ ext{√}11}$ and $0 < x < rac{ ext{π}}{4}$ - Scottish Highers Maths - Question 13 - 2018

Worked Solution & Example Answer:The right-angled triangle in the diagram is such that $ ext{sin } x = rac{2}{ ext{√}11}$ and $0 < x < rac{ ext{π}}{4}$ - Scottish Highers Maths - Question 13 - 2018

Find the exact value of: (i) sin 2x

Find the exact value of: (ii) cos 2x

By expressing sin 3x as sin(2x + x), find the exact value of sin 3x

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