Question 53

If $$x = 4 \cos A + 5 \sin A$$ and $$y = 4 \sin A - 5 \cos A$$, then the value of $$x^2 + y^2$$ is:

Solution

$$x = 4 \cos A + 5 \sin A$$

$$y = 4 \sin A - 5 \cos A$$

$$x^2 + y^2$$

= $$(4 \cos A + 5 \sin A)^2 + (4 \sin A - 5 \cos A)^2$$

($$(a + b)^2 = a^2 + b^2 + 2ab$$)

$$= (16 \cos^2 A + 25 \sin^2 A + 40\cos A\sin A) + (16 \sin^2 A + 25 \cos^2 A - 40\cos A\sin A)^2$$

($$ \sin^2 A + \cos^2 A = 1$$)

= 16 + 25 = 41

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