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The diagram below shows a right-angled triangle - Junior Cycle Mathematics - Question 11 - 2017

Question 11

The diagram below shows a right-angled triangle. One side has a length of $x$ units, and the hypotenuse is 3 units in length. One of the angles is 65°, as shown. (a... show full transcript

Worked Solution & Example Answer:The diagram below shows a right-angled triangle - Junior Cycle Mathematics - Question 11 - 2017

Step 1

Using the diagram, write sin 65° as a fraction in terms of x.

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Answer

From the triangle, the sine of an angle is defined as the ratio of the opposite side to the hypotenuse. Here, we have:

$\sin 65° = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{x}{3}$

Therefore, we can write:

$\sin 65° = \frac{x}{3}$

Step 2

Use a calculator to find the value of sin 65°.

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Answer

Using a calculator, we find:

$\sin 65° \approx 0.906$

Rounding to one decimal place, the value is:

$\sin 65° \approx 0.9$

Step 3

Use your answers from part (a) and part (b) to find the value of x.

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Answer

Substituting the value of sin 65° from part (b) into the equation from part (a):

$\frac{x}{3} = 0.9$

To isolate $x$ , multiply both sides by 3:

$x = 3 \times 0.9 = 2.7$

Thus, the value of $x$ is 2.7 units.

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