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

1. (i) Solve, for \(0 \leqslant x < \frac { \pi } { 2 }\), the equation

$$4 \sin x = \sec x$$ (ii) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$5 \sin \theta - 5 \cos \theta = 2$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. (a) Prove that

$$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.

1. (a) Prove

$$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } \equiv 2 \cot 2 \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } = 4$$ giving any solutions to one decimal place.

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

\section*{Solutions relying on calculator technology are not acceptable.} Given that the first three terms of a geometric series are $$12 \cos \theta \quad 5 + 2 \sin \theta \quad \text { and } \quad 6 \tan \theta$$

1. show that $$4 \sin ^ { 2 } \theta - 52 \sin \theta + 25 = 0$$ Given that \(\theta\) is an obtuse angle measured in radians,
2. solve the equation in part (a) to find the exact value of \(\theta\)
3. show that the sum to infinity of the series can be expressed in the form $$k ( 1 - \sqrt { 3 } )$$ where \(k\) is a constant to be found.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$2 \tan \theta \left( 8 \cos \theta + 23 \sin ^ { 2 } \theta \right) = 8 \sin 2 \theta \left( 1 + \tan ^ { 2 } \theta \right)$$ may be written as $$\sin 2 \theta \left( A \cos ^ { 2 } \theta + B \cos \theta + C \right) = 0$$ where \(A , B\) and \(C\) are constants to be found.
2. Hence, solve for \(360 ^ { \circ } \leqslant x \leqslant 540 ^ { \circ }\) $$2 \tan x \left( 8 \cos x + 23 \sin ^ { 2 } x \right) = 8 \sin 2 x \left( 1 + \tan ^ { 2 } x \right) \quad x \in \mathbb { R } \quad x \neq 450 ^ { \circ }$$

1. A student attempts to answer the following question:

Given that \(x\) is an obtuse angle, use algebra to prove by contradiction that $$\sin x - \cos x \geqslant 1$$ The student starts the proof with: Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\begin{aligned} & \Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1 \\ & \Rightarrow \ldots \end{aligned}$$ The start of the student's proof is reprinted below.
Complete the proof. Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\frac { 1 } { \operatorname { cosec } \theta - 1 } + \frac { 1 } { \operatorname { cosec } \theta + 1 } \equiv 2 \tan \theta \sec \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$
2. Hence solve, for \(0 < x < 90 ^ { \circ }\), the equation $$\frac { 1 } { \operatorname { cosec } 2 x - 1 } + \frac { 1 } { \operatorname { cosec } 2 x + 1 } = \cot 2 x \sec 2 x$$ Give each answer, in degrees, to one decimal place.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$
2. Hence solve, for \(- 90 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), the equation $$1 - \cos 3 x = \sin ^ { 2 } x$$

1. (a) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation

$$3 \sin ^ { 2 } x + \sin x + 8 = 9 \cos ^ { 2 } x$$ giving your answers to 2 decimal places.
(b) Hence find the smallest positive solution of the equation $$3 \sin ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) + \sin \left( 2 \theta - 30 ^ { \circ } \right) + 8 = 9 \cos ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right)$$ giving your answer to 2 decimal places.

4 In this question you must show detailed reasoning. Solve the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(2 \tan x + 1 = 4\)
2. \(5 \sin x - 1 = 2 \cos ^ { 2 } x\)

7

1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
2. Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.

5

1. Show that the equation \(2 \cos x \tan ^ { 2 } x = 3 ( 1 + \cos x )\) can be expressed in the form $$5 \cos ^ { 2 } x + 3 \cos x - 2 = 0$$ \section*{(b) In this question you must show detailed reasoning.} Hence solve the equation $$2 \cos 3 \theta \tan ^ { 2 } 3 \theta = 3 ( 1 + \cos 3 \theta ) ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(120 ^ { \circ }\), correct to \(\mathbf { 1 }\) decimal place where appropriate.

4 In this question you must show detailed reasoning.
Solve the equation \(6 \cos ^ { 2 } x + \sin x = 5\), giving all the roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

14 In this question you must show detailed reasoning.
Solve the equation \(5 - \cos \theta - 6 \sin ^ { 2 } \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). Turn over for question 15

8 In this question you must show detailed reasoning.
Solve the equation \(3 \cos \theta + 8 \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers correct to the nearest degree.

12 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: Let \(\arcsin x = \theta\) ] \section*{END OF QUESTION PAPER}

6

1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
   Section B (75 marks)

15 A projectile is launched from a point on level ground with an initial velocity \(u\) at an angle \(\theta\) above the horizontal.

1. Show that the range of the projectile is given by \(\frac { 2 u ^ { 2 } \sin \theta \cos \theta } { g }\).
2. Determine the set of values of \(\theta\) for which the maximum height of the projectile is greater than the range, where \(\theta\) is an acute angle. Give your answer in degrees.

4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).

12

1. Show that \(\cos x = \sin \left( x + \frac { \pi } { 2 } \right)\).
2. Hence show that \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) gives the approximation \(\cos x \approx \frac { \pi ^ { 2 } - 4 x ^ { 2 } } { \pi ^ { 2 } + x ^ { 2 } }\), as stated in line 31. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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1 In this question you must show detailed reasoning.
The obtuse angle \(\theta\) is such that \(\sin \theta = \frac { 2 } { \sqrt { 13 } }\).
Find the exact value of \(\cos \theta\).

10

1. You are given that \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = x ^ { 6 } + 3 x ^ { 4 } y ^ { 2 } + 3 x ^ { 2 } y ^ { 4 } + y ^ { 6 }\).
   Hence, or otherwise, prove that \(\sin ^ { 6 } \theta + \cos ^ { 6 } \theta = 1 - \frac { 3 } { 4 } \sin ^ { 2 } 2 \theta\) for all values of \(\theta\).
2. Use the result from part (a) to determine the minimum value of \(\sin ^ { 6 } \theta + \cos ^ { 6 } \theta\). The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

7 Prove that \(\sin 8 \theta \tan 4 \theta + \cos 8 \theta = 1\).

6

1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
   1. \(y = 2 \sin x\);
   2. \(y = - \sin x\);
   3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

9

1. Given that $$\frac { 3 + \sin ^ { 2 } \theta } { \cos \theta - 2 } = 3 \cos \theta$$ show that $$\cos \theta = - \frac { 1 } { 2 }$$
2. Hence solve the equation $$\frac { 3 + \sin ^ { 2 } 3 x } { \cos 3 x - 2 } = 3 \cos 3 x$$ giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).