1.5 Trigonometry

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Question 8

A high-precision sensor measures the output voltage $V\,$ of a microscopic resonator as a function of its deflection angle $θ\theta$ (in radians). The sensor output is modeled by the equation:

V(\theta) = 3\cos(2\theta) - 3\cos(2\theta)\cos(6\theta)

Show that for small values of $θ\theta$ , $V(θ)≈54θ2V(\theta) \approx 54\theta^2$ .

[4]

The energy $E\,$ dissipated during an oscillation cycle is given by $dθ\displaystyle E = \int_{0}^{0.1} \sqrt{\frac{1}{54}V(\theta)} \, d\theta$ . Show that the energy $E\,$ can be approximated by $\approx 2^m \times 5^n$ , where $m\,$ and $n\,$ are integers to be determined.

[5]

ci.

Explain why $dθ\int_{12.6}^{12.7} \theta \, d\theta$ is not a suitable approximation for $dθ\displaystyle \int_{12.6}^{12.7} \sqrt{\frac{1}{54}V(\theta)} \, d\theta$ .

[1]

cii.

Explain how $dθ\displaystyle \int_{12.6}^{12.7} \sqrt{\frac{1}{54}V(\theta)} \, d\theta$ may be approximated by $dθ\int_{a}^{b} \theta \, d\theta$ for suitable values of $a\,$ and $b$ .

[2]

1.5 Trigonometry

Question 8

Markscheme

1.5 Trigonometry

Question 8

Markscheme

1.5 Trigonometry

Question 8

1.5 Trigonometry Questions

1.5 Trigonometry

Question 8

1.5 Trigonometry Questions