Exam-Style Question on Trigonometry

A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 412. This question is similar to one that appeared on an IB Standard paper in 2001. The use of a calculator is allowed.

The diagram shown the origin O (0,0) and a fixed point A(10,2). The point P moves along the horizontal line $$y = 8$$.

(a) Show that $$PA = \sqrt{x^2 - 20x + 136}$$

(b) Write down a similar expression for $$OP$$ in terms of $$x$$

(c) Hence, show that:

$$\cos O\hat{P}A = \frac{x^2-10x+48}{\sqrt{(x^2 + 64)(x^2 -20x +136)}}$$

Let this expression for the cosine of $$O\hat{P}A$$ be defined as function $$f$$.

(d) Find the size of angle $$O\hat{P}A$$ in degrees when $$x=5$$.

(e) Find two positive values of $$x$$ such that $$O\hat{P}A = 60^o$$.

(f) Consider the equation $$f(x) = 1$$. Explain, in terms of the positions of the points O, A and P, why this equation has a solution.

(g) Find the exact solution of $$f(x) = 1$$.

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A GeoGebra interactive in which P can be dragged along the red line to show how the angle changes is available here. If you would like to interact with the graph mentioned in this question you will find it waiting for you on the Graph Plotter.

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