Solve trigonometric equation with approximate values

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1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0 .$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\).
   2. Express \(\frac { y + 3 } { ( y + 1 ) ( y + 2 ) } - \frac { y + 1 } { ( y + 2 ) ( y + 3 ) }\) as a single fraction in its simplest form.
   3. Given that \(2 \sin 2 \theta = \cos 2 \theta\),
3. show that \(\tan 2 \theta = 0.5\).
4. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }\).
   4. \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c\), where \(c\) is a constant. Given that \(\mathrm { f } ( 4 ) = 0\),
5. find the value of \(c\),
6. factorise \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
   (c Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(\mathrm { f } ( x ) = 0\).
   5. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13c2bf9f-f87a-420c-8cdc-9deb688112ae-3_538_618_283_749}
   \end{figure} Figure 1 shows the sector \(O A B\) of a circle of radius \(r \mathrm {~cm}\). The area of the sector is \(15 \mathrm {~cm} ^ { 2 }\) and \(\angle A O B = 1.5\) radians.
7. Prove that \(r = 2 \sqrt { } 5\).
8. Find, in cm , the perimeter of the sector \(O A B\). The segment \(R\), shaded in Fig 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
9. Calculate, to 3 decimal places, the area of \(R\).
   6. The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find
10. the common ratio of the series,
11. the first term of the series,
12. the sum to infinity of the series.
13. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series.
    7. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180 .$$
14. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis
15. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
16. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\)

14. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]
\includegraphics[max width=\textwidth, alt={}, center]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-10_835_974_379_438} Figure 1 shows a gardener's design for the shape of a flower bed with perimeter \(A B C D\).
\(A D\) is an arc of a circle with centre \(O\) and radius 5 m .
\(B C\) is an arc of a circle with centre \(O\) and radius 7 m .
\(O A B\) and \(O D C\) are straight lines and the size of \(\angle A O D\) is \(\theta\) radians.

1. Find, in terms of \(\theta\), an expression for the area of the flower bed. Given that the area of the flower bed is \(15 \mathrm {~m} ^ { 2 }\),
2. show that \(\theta = 1.25\),
3. calculate, in m , the perimeter of the flower bed.
   (3) The gardener now decides to replace arc \(A D\) with the straight line \(A D\).
4. Find, to the nearest cm , the reduction in the perimeter of the flower bed.
   (2)