Find value where max/min occurs

7

1. Express \(5 \cos \theta - 12 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(5 \cos \theta - 12 \sin \theta = 8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
3. Find the greatest possible value of $$7 + 5 \cos \frac { 1 } { 2 } \phi - 12 \sin \frac { 1 } { 2 } \phi$$ as \(\phi\) varies, and determine the smallest positive value of \(\phi\) for which this greatest value occurs.
   [0pt] [4]

6

1. By first expanding \(\cos \left( x - 60 ^ { \circ } \right)\), show that the expression $$2 \cos \left( x - 60 ^ { \circ } \right) + \cos x$$ can be written in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) for which \(2 \cos \left( x - 60 ^ { \circ } \right) + \cos x\) takes its least possible value.

1. (a) Express \(12 \sin x - 5 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) in radians, to 3 decimal places.

The function g is defined by $$g ( \theta ) = 10 + 12 \sin \left( 2 \theta - \frac { \pi } { 6 } \right) - 5 \cos \left( 2 \theta - \frac { \pi } { 6 } \right) \quad \theta > 0$$ Find
(b) (i) the minimum value of \(\mathrm { g } ( \theta )\) (ii) the smallest value of \(\theta\) at which the minimum value occurs. The function h is defined by $$\mathrm { h } ( \beta ) = 10 - ( 12 \sin \beta - 5 \cos \beta ) ^ { 2 }$$ (c) Find the range of h .

1. (a) Express \(35 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$70 \sin x - 24 \cos x = 37$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$y = \frac { 7000 } { 31 + ( 35 \sin x - 12 \cos x ) ^ { 2 } } , \quad x > 0$$ (c) Use your answer to part (a) to calculate

1. the minimum value of \(y\),
2. the smallest value of \(x , x > 0\), at which this minimum value occurs.

1. (a) Express \(6 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).

Give the value of \(\alpha\) to 3 decimal places.
(b) $$\mathrm { p } ( \theta ) = \frac { 4 } { 12 + 6 \cos \theta + 8 \sin \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Calculate

1. the maximum value of \(\mathrm { p } ( \theta )\),
2. the value of \(\theta\) at which the maximum occurs.

9. (a) Express \(2 \sin \theta - 4 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(\alpha\) to 3 decimal places. $$H ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$ Find
(b) (i) the maximum value of \(\mathrm { H } ( \theta )\),
(ii) the smallest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum value occurs. Find
(c) (i) the minimum value of \(\mathrm { H } ( \theta )\),
(ii) the largest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this minimum value occurs.

1. (a) Express \(\sin \theta - 2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and the value of \(\alpha\), in radians, to 3 decimal places. $$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$ (b) Find

1. the maximum value of \(\mathrm { M } ( \theta )\),
2. the smallest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { M } ( \theta )\) occurs. $$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$ (c) Find
3. the maximum value of \(\mathrm { N } ( \theta )\),
4. the largest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { N } ( \theta )\) occurs.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

   END

5 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).

7 Fig. 1 shows part of the graph of \(y = \sin x \quad \sqrt { 3 } \cos x\). \begin{figure}[h] \end{figure} Express \(\quad \sqrt { } \quad\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi\).
Hence write down the exact coordinates of the turning point P .

2 Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 \sin x + 2 \cos x , 0 \leqslant x \leqslant \pi$$

8

1. Express \(4 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence
   (a) solve the equation \(4 \cos \theta - 2 \sin \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
   (b) determine the greatest and least values of $$25 - ( 4 \cos \theta - 2 \sin \theta ) ^ { 2 }$$ as \(\theta\) varies, and, in each case, find the smallest positive value of \(\theta\) for which that value occurs.

3 Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 \sin x + 2 \cos x , 0 \leqslant x \leqslant \pi .$$

6. a. Express \(4 \sin x - 5 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\), and give the value of \(\alpha\), in degrees, to 2 decimal places. $$T = \frac { 8400 } { 19 + ( 4 \sin x - 5 \cos x ) ^ { 2 } } , x > 0$$ b. Use your answer to part \(a\) to calculate
i. the minimum value of \(T\).
ii. the smallest value of \(x , x > 0\), at which this minimum value occurs.

1

1. Express \(2 \sin x + 5 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. 1. Write down the maximum value of \(2 \sin x + 5 \cos x\).
   2. Find the value of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) at which this maximum occurs, giving the value of \(x\) to the nearest \(0.1 ^ { \circ }\).

3

1. 1. Express \(3 \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
      (3 marks)
   2. Hence find the minimum value of \(3 \cos x + 2 \sin x\) and the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) where the minimum occurs. Give your value of \(x\) to the nearest \(0.1 ^ { \circ }\).
2. 1. Show that \(\cot x - \sin 2 x = \cot x \cos 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
   2. Hence, or otherwise, solve the equation $$\cot x - \sin 2 x = 0$$ in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).

2

1. Express \(\sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) in radians to two decimal places.
2. Hence:
   1. write down the minimum value of \(\sin x - 3 \cos x\);
   2. find the value of \(x\) in the interval \(0 < x < 2 \pi\) at which this minimum value occurs, giving your value of \(x\) in radians to two decimal places.

1 Fig. 1 shows part of the graph of \(y = \sin x - \sqrt { 3 } \cos x\). \begin{figure}[h] \end{figure} Express \(\sin x - \sqrt { 3 } \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi\).
Hence write down the exact coordinates of the turning point P .

7 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).