1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1

7. Given that $$x = 6 \sin ^ { 2 } 2 y \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { A \sqrt { \left( B x - x ^ { 2 } \right) } }$$ where \(A\) and \(B\) are integers to be found.
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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. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that the equation $$2 \sin \theta \left( 3 \cot ^ { 2 } 2 \theta - 7 \right) = 13 \sec \theta$$ can be written as $$3 \operatorname { cosec } ^ { 2 } 2 \theta - 13 \operatorname { cosec } 2 \theta - 10 = 0$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\), the equation $$2 \sin \theta \left( 3 \cot ^ { 2 } 2 \theta - 7 \right) = 13 \sec \theta$$ giving your answers to 3 significant figures.

9. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows the curve with equation $$y = \frac { 1 + 2 \cos x } { 1 + \sin x } \quad - \frac { \pi } { 2 } < x < \frac { 3 \pi } { 2 }$$ The point \(M\), shown in Figure 5, is the minimum point on the curve.

1. Show that the \(x\) coordinate of \(M\) is a solution of the equation $$2 \sin x + \cos x = - 2$$
2. Hence find, to 3 significant figures, the \(x\) coordinate of \(M\).

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$

6. (i) The curve \(C _ { 1 }\) has equation $$y = 3 \ln \left( x ^ { 2 } - 5 \right) - 4 x ^ { 2 } + 15 \quad x > \sqrt { 5 }$$ Show that \(C _ { 1 }\) has a stationary point at \(x = \frac { \sqrt { p } } { 2 }\) where \(p\) is a constant to be found.
(ii) A different curve \(C _ { 2 }\) has equation $$y = 4 x - 12 \sin ^ { 2 } x$$

1. Show that, for this curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = A + B \sin 2 x$$ where \(A\) and \(B\) are constants to be found.
2. Hence, state the maximum gradient of this curve.

6. $$y = \frac { 2 + 3 \sin x } { \cos x + \sin x }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a \tan x + b \sec x + c } { \sec x + 2 \sin x }$$ where \(a , b\) and \(c\) are integers to be found.

9. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Given that \(\cos 2 \theta - \sin 3 \theta \neq 0\)

1. prove that $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } \equiv \frac { 1 + \sin \theta } { 1 - 2 \sin \theta - 4 \sin ^ { 2 } \theta }$$
2. Hence solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } = 2 \operatorname { cosec } \theta$$ Give your answers to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-28_2257_52_309_1983}

1. (a) Prove that

$$2 \operatorname { cosec } ^ { 2 } 2 \theta ( 1 - \cos 2 \theta ) \equiv 1 + \tan ^ { 2 } \theta$$ (b) Hence solve for \(0 < x < 360 ^ { \circ }\), where \(x \neq ( 90 n ) ^ { \circ } , n \in \mathbb { N }\), the equation $$2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = 4 + 3 \sec x$$ giving your answers to one decimal place.
(Solutions relying entirely on calculator technology are not acceptable.)

9. (a) Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\) $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.

8. (a) Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)

1. \(2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3\)
2. \(\tan \theta + \cot \theta = 5\) Give your answers to 3 significant figures.

2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (a) Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (b) Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.

8. $$f ( \theta ) = 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta$$

1. Show that \(\mathrm { f } ( \theta ) = a + b \cos 2 \theta\), where \(a\) and \(b\) are integers which should be found.
2. Using your answer to part (a) and integration by parts, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta ^ { 2 } \mathrm { f } ( \theta ) \mathrm { d } \theta$$

5. The angle \(x\) and the angle \(y\) are such that $$\tan x = m \text { and } 4 \tan y = 8 m + 5$$ where \(m\) is a constant.
Given that \(16 \sec ^ { 2 } x + 16 \sec ^ { 2 } y = 537\)

1. find the two possible values of \(m\). Given that the angle \(x\) and the angle \(y\) are acute, find the exact value of
2. \(\sin x\)
3. \(\cot y\)

7. (a) Use the substitution \(t = \tan x\) to show that the equation $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$3 t ^ { 4 } + 8 t ^ { 2 } - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ Give each answer in terms of \(\pi\). You must make your method clear.

1. (a) Prove that

$$\frac { 1 - \cos 2 x } { 1 + \cos 2 x } \equiv \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$\frac { 2 - 2 \cos 2 \theta } { 1 + \cos 2 \theta } - 2 = 7 \sec \theta$$ Give your answers in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

12. (a) Show that $$\cot x - \tan x \equiv 2 \cot 2 x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 + \cot \left( \theta - 15 ^ { \circ } \right) - \tan \left( \theta - 15 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

4. (a) Prove that $$\frac { 1 - \cos 2 x } { \sin 2 x } \equiv \tan x , \quad x \neq \frac { n \pi } { 2 }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$3 \sec ^ { 2 } \theta - 7 = \frac { 1 - \cos 2 \theta } { \sin 2 \theta }$$ Give your answers in radians to 3 decimal places, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

3. Given \(\cos \theta ^ { \circ } = p\), where \(p\) is a constant and \(\theta ^ { \circ }\) is acute use standard trigonometric identities to find, in terms of \(p\),

1. \(\sec \theta ^ { \circ }\)
2. \(\sin ( \theta - 90 ) ^ { \circ }\)
3. \(\sin 2 \theta ^ { \circ }\) Write each answer in its simplest form.

7. A curve has equation $$y = \ln ( 1 - \cos 2 x ) , \quad x \in \mathbb { R } , 0 < x < \pi$$ Show that

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k \cot x\), where \(k\) is a constant to be found. Hence find the exact coordinates of the point on the curve where
2. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { 3 }\)

1. (a) Show that

$$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x , \quad x \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve for \(0 \leqslant \theta \leqslant \pi\) $$\operatorname { cosec } \left( 3 \theta + \frac { \pi } { 3 } \right) + \cot \left( 3 \theta + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 3 } }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (a) Show that

1. \(\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }\),
2. \(\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }\).
   (b) Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$ (c) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of \(\pi\).

1. 1. Prove that

$$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$

1. express arcsin \(x\) in terms of \(y\).
2. Hence evaluate \(\arccos x + \arcsin x\). Give your answer in terms of \(\pi\).

6. (a) Use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to obtain an expression for \(\cos 3 x\) in terms of powers of \(\cos x\) only.
(b) (i) Prove that $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$ (ii) Hence find, for \(0 < x < 2 \pi\), all the solutions of $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$