1.05p Proof involving trig: functions and identities

2

1. Show that \(\cos ( \alpha + \beta ) = \frac { 1 - \tan \alpha \tan \beta } { \sec \alpha \sec \beta }\).
2. Hence show that \(\cos 2 \alpha = \frac { 1 - \tan ^ { 2 } \alpha } { 1 + \tan ^ { 2 } \alpha }\).
3. Hence or otherwise solve the equation \(\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4 Prove that \(\sec ^ { 2 } \theta + \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta \operatorname { cosec } ^ { 2 } \theta\).

9

1. By first expanding \(\cos ( 2 \theta + \theta )\), prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
2. Hence prove that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
3. Show that the only solutions of the equation $$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$ are odd multiples of \(90 ^ { \circ }\).

9

1. Prove that \(\frac { \sin ( \theta - \alpha ) + 3 \sin \theta + \sin ( \theta + \alpha ) } { \cos ( \theta - \alpha ) + 3 \cos \theta + \cos ( \theta + \alpha ) } \equiv \tan \theta\) for all values of \(\alpha\).
2. Find the exact value of \(\frac { 4 \sin 149 ^ { \circ } + 12 \sin 150 ^ { \circ } + 4 \sin 151 ^ { \circ } } { 3 \cos 149 ^ { \circ } + 9 \cos 150 ^ { \circ } + 3 \cos 151 ^ { \circ } }\).
3. It is given that \(k\) is a positive constant. Solve, for \(0 ^ { \circ } < \theta < 60 ^ { \circ }\) and in terms of \(k\), the equation $$\frac { \sin \left( 6 \theta - 15 ^ { \circ } \right) + 3 \sin 6 \theta + \sin \left( 6 \theta + 15 ^ { \circ } \right) } { \cos \left( 6 \theta - 15 ^ { \circ } \right) + 3 \cos 6 \theta + \cos \left( 6 \theta + 15 ^ { \circ } \right) } = k .$$

4 Prove that \(\cot \beta - \cot \alpha = \frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta }\).

5 Show that \(\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta\).

10

1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

9. $$f ( x ) = \frac { 50 x ^ { 2 } + 38 x + 9 } { ( 5 x + 2 ) ^ { 2 } ( 1 - 2 x ) } \quad x \neq - \frac { 2 } { 5 } \quad x \neq \frac { 1 } { 2 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$\frac { A } { 5 x + 2 } + \frac { B } { ( 5 x + 2 ) ^ { 2 } } + \frac { C } { 1 - 2 x }$$ where \(A\), \(B\) and \(C\) are constants

1. 1. find the value of \(B\) and the value of \(C\)
   2. show that \(A = 0\)
2. 1. Use binomial expansions to show that, in ascending powers of \(x\) $$f ( x ) = p + q x + r x ^ { 2 } + \ldots$$ where \(p , q\) and \(r\) are simplified fractions to be found.
   2. Find the range of values of \(x\) for which this expansion is valid.

8. $$f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30 , \quad x > 2.5$$

1. Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [3.5,4] A student takes 4 as the first approximation to \(\alpha\).
   Given \(\mathrm { f } ( 4 ) = 3.099\) and \(\mathrm { f } ^ { \prime } ( 4 ) = 16.67\) to 4 significant figures,
2. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures.
3. Show that \(\alpha\) is the only root of \(\mathrm { f } ( x ) = 0\)

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.

3 In this question you must show detailed reasoning.

1. Solve the equation \(4 \sin ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
2. Prove that \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).

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. \(y = \sin \left( x - 30 ^ { \circ } \right)\).
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 \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

1. 1. Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to prove that

$$\cot x + \tan \left( \frac { x } { 2 } \right) = \operatorname { cosec } x \quad x \neq n \pi , n \in \mathbb { Z }$$ (ii) \begin{figure}[h] \end{figure} An engineer models the vertical height above the ground of the tip of one blade of a wind turbine, shown in Figure 1. The ground is assumed to be horizontal. The vertical height of the tip of the blade above the ground, \(H\) metres, at time \(x\) seconds after the wind turbine has reached its constant operating speed, is modelled by the equation $$H = 90 - 30 \cos ( 120 x ) ^ { \circ } - 40 \sin ( 120 x ) ^ { \circ }$$

1. Show that \(H = 60\) when \(x = 0\) Using the substitution \(t = \tan ( 60 x ) ^ { \circ }\)
2. show that equation (I) can be rewritten as $$H = \frac { 120 t ^ { 2 } - 80 t + 60 } { 1 + t ^ { 2 } }$$
3. Hence find, according to the model, the value of \(x\) when the tip of the blade is 100 m above the ground for the first time after the wind turbine has reached its constant operating speed.

1. (a) Use \(t = \tan \frac { \theta } { 2 }\) to show that, where both sides are defined

$$\frac { 29 - 21 \sec \theta } { 20 - 21 \tan \theta } \equiv \frac { 5 t + 2 } { 2 t + 5 }$$ (b) Hence, again using \(t = \tan \frac { \theta } { 2 }\), prove that, where both sides are defined $$\frac { 20 + 21 \tan \theta } { 29 + 21 \sec \theta } \equiv \frac { 29 - 21 \sec \theta } { 20 - 21 \tan \theta }$$

1. (a) Given that \(t = \tan \frac { X } { 2 }\) prove that

$$\cos x \equiv \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (b) Show that the equation $$3 \tan x - 10 \cos x = 10$$ can be written in the form $$( t + 2 ) \left( a t ^ { 2 } + b t + c \right) = 0$$ where \(t = \tan \frac { X } { 2 }\) and \(a , b\) and \(c\) are integers to be determined.
(c) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \tan x - 10 \cos x = 10$$

1. (a) Use the substitution \(\mathrm { t } = \tan \left( \frac { \mathrm { x } } { 2 } \right)\) to show that

$$\sec x - \tan x \equiv \frac { 1 - t } { 1 + t } \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Use the substitution \(\mathrm { t } = \tan \left( \frac { \mathrm { x } } { 2 } \right)\) and the answer to part (a) to prove that $$\frac { 1 - \sin x } { 1 + \sin x } \equiv ( \sec x - \tan x ) ^ { 2 } \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ \section*{Q uestion 1 continued}

7

1. Use the identity $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$ to express \(\tan 2 x\) in terms of \(\tan x\).
2. Show that $$2 - 2 \tan x - \frac { 2 \tan x } { \tan 2 x } = ( 1 - \tan x ) ^ { 2 }$$ for all values of \(x , \tan 2 x \neq 0\).

7

1. 1. Express \(6 \sin \theta + 8 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(6 \sin 2 x + 8 \cos 2 x = 7\), giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. 1. Prove the identity \(\frac { \sin 2 x } { 1 - \cos 2 x } = \frac { 1 } { \tan x }\).
   2. Hence solve the equation $$\frac { \sin 2 x } { 1 - \cos 2 x } = \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Solve the equation $$5 \sin 2 x + 3 \cos x = 0$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) to the nearest \(0.1 ^ { \circ }\), where appropriate.
3. Given that \(\sin 2 x + \cos 2 x = 1 + \sin x\) and \(\sin x \neq 0\), show that \(2 ( \cos x - \sin x ) = 1\).

6

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Using the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) :
   1. express \(\cos 2 x\) in terms of \(\sin x\) and \(\cos x\);
   2. show, by writing \(3 x\) as \(( 2 x + x )\), that $$\cos 3 x = 4 \cos ^ { 3 } x - 3 \cos x$$
3. Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }\).

3.(i)Determine the value of \(k\) such that $$\arctan \frac { 1 } { 2 } - \arctan \frac { 1 } { 3 } = \arctan k$$ (ii)(a)Prove that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$ Given that \(a = \cos 20 ^ { \circ }\) (b)write down,in terms of \(a\) ,an expression for \(\cos 40 ^ { \circ }\) (c)determine,in terms of \(a\) ,a simplified expression for \(\cos 80 ^ { \circ }\) (d)Use part(a)to show that $$4 a ^ { 3 } - 3 a = \frac { 1 } { 2 }$$ (e)Hence,or otherwise,show that $$\cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 80 ^ { \circ } = \frac { 1 } { 8 }$$

10

1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).

10

1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
2. Hence solve the equation \(\cot \left( \theta + \frac { \neq } { 4 } \right) + \frac { \sin \left( \theta + \frac { \neq } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \neq } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).