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Consider the triangle shown - HSC - SSCE Mathematics Standard - Question 16 - 2020 - Paper 1

Question 16

Consider the triangle shown. (a) Find the value of θ, correct to the nearest degree. (b) Find the value of x, correct to one decimal place.

Worked Solution & Example Answer:Consider the triangle shown - HSC - SSCE Mathematics Standard - Question 16 - 2020 - Paper 1

Step 1

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Answer

To find the angle θ in the triangle, we can use the tangent function since we have the lengths of the opposite side (8) and the adjacent side (10).

Using the formula for tangent:

$\tan(θ) = \frac{\text{opposite}}{\text{adjacent}}$

This gives us:

$\tan(θ) = \frac{8}{10}$

Calculating θ using the arctangent function:

$θ = \arctan\left(\frac{8}{10}\right)$

Using a calculator, we find:

$θ ≈ 39°$

Thus, the value of θ is 39° (nearest degree).

Step 2

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Answer

To find the value of x (the length of the adjacent side), we can use the Pythagorean theorem. In a right triangle:

$a^2 + b^2 = c^2$

where c is the hypotenuse (10), a is the opposite side (8), and b is the adjacent side (x).

Substituting the known values:

$8^2 + x^2 = 10^2$

Calculating:

$64 + x^2 = 100$

Rearranging gives:

$x^2 = 100 - 64$

$x^2 = 36$

Taking the square root:

$x = \sqrt{36} = 6$

Therefore, the value of x is 6.0 (correct to one decimal place).

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