A heating engineer's invoice for Grace's household boiler installation shows a bracket fitted at \(45^\circ\). Grace checks part of the engineer's calculati...
Question 1 Report
A heating engineer's invoice for Grace's household boiler installation shows a bracket fitted at \(45^\circ\). Grace checks part of the engineer's calculation by hand, without a calculator.
- Write down the exact value of \(\sin 45^\circ\). (1)
- State the exact value of \(\cos 45^\circ\). (1)
- Hence show that \(\sin^2 45^\circ + \cos^2 45^\circ = 1\). (1)
Answer Details
This question tests recall of the exact value of \( \sin 45^\circ \) and \( \cos 45^\circ \), and using them to verify the Pythagorean identity for a specific angle, all without a calculator.
(a) In a right-angled isosceles triangle with both shorter sides of length 1, the hypotenuse has length \( \sqrt{2} \):
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \] [1 mark]
(b) By the symmetry of the same triangle:
\[ \cos 45^\circ = \frac{\sqrt{2}}{2} \] [1 mark]
(c) Substituting these exact values:
\[ \sin^2 45^\circ + \cos^2 45^\circ = \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1 \] [1 mark]
This confirms, for the special case \( 45^\circ \), the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), which follows from Pythagoras' theorem applied to the right-angled triangle used to define sine and cosine.
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