Mathematics — AI-generated practice question

$\sin 45° = \cos 45° = \dfrac{1}{\sqrt{2}}$

In a right triangle with one angle of 45°, the other acute angle is also 45°. This makes the triangle isosceles, so the two legs are equal (BC = AB = $a$). By Pythagoras theorem, hypotenuse $= a\sqrt{2}$.

Since both the opposite and adjacent sides are equal, $\sin 45° = \dfrac{a}{a\sqrt{2}} = \dfrac{1}{\sqrt{2}}$ and $\cos 45° = \dfrac{a}{a\sqrt{2}} = \dfrac{1}{\sqrt{2}}$, hence they are equal.

Source: Chapter 8, Section 8.3

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• The key insight is that a 45°-45°-90° triangle is isosceles — both legs are equal, so the numerators of sin and cos are the same, giving equal values.
• Write the derivation briefly; don't just state the value.
• For 2 marks: state the value (1 mark) + give the reason using the isosceles triangle argument (1 mark).