## Exam-Style Question on Trigonometry## A mathematics exam-style question with a worked solution that can be revealed gradually |

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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