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What is the value of $ an heta$? A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2018 - Paper 1

Question 5

What is the value of $ an heta$? A. $rac{1}{8}$ B. $rac{1}{7}$ C. $rac{1}{2}$ D. $rac{4}{7}$ What is the value of $$ ext{lim}_{x o 0} rac{ ext{sin}(3x) ext... show full transcript

Worked Solution & Example Answer:What is the value of $ an heta$? A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2018 - Paper 1

Step 1

What is the value of $ an heta$?

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Answer

To find the value of $an heta$ , we can assume that there is a right triangle where the opposite side has length 1, and the adjacent side has length 8 for option A, therefore:

$an heta = rac{ ext{opposite}}{ ext{adjacent}} = rac{1}{8}$

So, $an heta$ equals $rac{1}{8}$ .

Step 2

What is the value of $$ ext{lim}_{x o 0} rac{ ext{sin}(3x) ext{cos}(3x)}{12x}$$?

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Answer

To evaluate this limit, we can use the fact that:

$ext{sin}(x) ext{cos}(x) = rac{1}{2} ext{sin}(2x)$

Thus, we can rewrite the limit as:

$ext{lim}_{x o 0} rac{ rac{1}{2} ext{sin}(6x)}{12x} = rac{1}{24} ext{lim}_{x o 0} rac{ ext{sin}(6x)}{x}$

Using the limit property $ext{lim}_{x o 0} rac{ ext{sin}(kx)}{x} = k$ , we find:

$ext{lim}_{x o 0} rac{ ext{sin}(6x)}{x} = 6$

Thus,

$ext{lim}_{x o 0} rac{ ext{sin}(3x) ext{cos}(3x)}{12x} = rac{1}{24} imes 6 = rac{1}{4}$

So the answer is $rac{1}{4}$ .

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