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The diagram shows a right-angled triangle, with the angle A marked - Junior Cycle Mathematics - Question b - 2016

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The diagram shows a right-angled triangle, with the angle A marked. Given that cos A = sin A, show that this triangle must be isosceles.

Worked Solution & Example Answer:The diagram shows a right-angled triangle, with the angle A marked - Junior Cycle Mathematics - Question b - 2016

Step 1

Show that cos A = sin A

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Answer

We start with the trigonometric identities:

cos⁡A=adjacenthypotenusesin⁡A=oppositehypotenuse\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \\ \sin A = \frac{\text{opposite}}{\text{hypotenuse}}

Setting them equal gives:

adjacenthypotenuse=oppositehypotenuse\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\text{opposite}}{\text{hypotenuse}}

Simplifying, we have:

adjacent=opposite\text{adjacent} = \text{opposite}

This indicates that the lengths of the two legs of the triangle are equal, thus proving that the triangle is isosceles.

Step 2

Find the size of the smallest angle in this triangle.

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Answer

Given a right-angled triangle with sides of lengths 7 cm, 24 cm, and 25 cm, we will use the sine function to find the smallest angle. Let angle A be opposite to the side of 7 cm:

sin⁡A=oppositehypotenuse=725\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{25}

Then, applying the arcsine function:

A=sin⁡−1(725)≈16.3∘A = \sin^{-1}\left(\frac{7}{25}\right) \approx 16.3^{\circ}

Therefore, the size of the smallest angle A, correct to one decimal place, is:

16.3°.

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