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

8

1. Solve the equation \(\tan \left( x + 52 ^ { \circ } \right) = \tan 22 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. 1. Show that the equation $$3 \tan \theta = \frac { 8 } { \sin \theta }$$ can be written as $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
   2. Find the value of \(\cos \theta\) that satisfies the equation $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
   3. Hence solve the equation $$3 \tan 2 x = \frac { 8 } { \sin 2 x }$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

9

1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
2. 1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
   2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)

8

1. Given that \(2 \sin \theta = 7 \cos \theta\), find the value of \(\tan \theta\).
2. 1. Use an appropriate identity to show that the equation $$6 \sin ^ { 2 } x = 4 + \cos x$$ can be written as $$6 \cos ^ { 2 } x + \cos x - 2 = 0$$
   2. Hence solve the equation \(6 \sin ^ { 2 } x = 4 + \cos x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving your answers to the nearest degree.

9

1. Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   (2 marks)
2. Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
3. 1. Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
   2. Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.

8

1. 1. Show that the equation $$4 \tan \theta \sin \theta = 15$$ can be written as $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ (1 mark)
   2. Use an appropriate identity to show that the equation $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ can be written as $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$
2. 1. Solve the equation \(4 c ^ { 2 } + 15 c - 4 = 0\).
   2. Hence explain why the only value of \(\cos \theta\) which satisfies the equation $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$ is \(\cos \theta = \frac { 1 } { 4 }\).
   3. Hence solve the equation \(4 \tan \theta \sin \theta = 15\) giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Write down all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\) for which $$4 \tan 4 x \sin 4 x = 15$$ giving your answers to the nearest degree.

8

1. Describe the single geometrical transformation by which the curve with equation \(y = \tan \frac { 1 } { 2 } x\) can be obtained from the curve \(y = \tan x\).
2. Solve the equation \(\tan \frac { 1 } { 2 } x = 3\) in the interval \(\mathbf { 0 } < \boldsymbol { x } < \mathbf { 4 } \boldsymbol { \pi }\), giving your answers in radians to three significant figures.
3. Solve the equation $$\cos \theta ( \sin \theta - 3 \cos \theta ) = 0$$ in the interval \(0 < \theta < 2 \pi\), giving your answers in radians to three significant figures.
   (5 marks)

7

1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leqslant x \leqslant 2 \pi\). State the values of the intercepts with the coordinate axes.
2. 1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that \(\cos \theta = \frac { 1 } { 2 }\).
   2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval \(0 \leqslant x \leqslant \pi\), giving your answers in radians to three significant figures.

8 Prove that, for all values of \(x\), the value of the expression $$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$ is an integer and state its value.

2 The triangle \(A B C\), shown in the diagram, is such that \(A B = 26 \mathrm {~cm}\) and \(B C = 31.5 \mathrm {~cm}\). The acute angle \(A B C\) is \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\).

1. Calculate the area of triangle \(A B C\).
2. Find the exact value of \(\cos \theta\).
3. Calculate the length of \(A C\).

7 It is given that \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\).

1. Find the possible values of \(\tan \theta\).
2. Hence solve the equation \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\), giving all solutions for \(\theta\), in degrees, in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

9

1. 1. On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   2. Solve the equation \(\tan x = - 1\), giving all values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. 1. Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
   2. Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}

7

1. Given that \(\frac { \cos ^ { 2 } x + 4 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 7\), show that \(\tan ^ { 2 } x = \frac { 3 } { 2 }\).
2. Hence solve the equation \(\frac { \cos ^ { 2 } 2 \theta + 4 \sin ^ { 2 } 2 \theta } { 1 - \sin ^ { 2 } 2 \theta } = 7\) in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), giving your values of \(\theta\) to the nearest degree.
   [0pt] [4 marks]

6

1. Solve the equation \(\sin ( x + 0.7 ) = 0.6\) in the interval \(- \pi < x < \pi\), giving your answers in radians to two significant figures.
2. It is given that \(5 \cos ^ { 2 } \theta - \cos \theta = \sin ^ { 2 } \theta\).
   1. By forming and solving a suitable quadratic equation, find the possible values of \(\cos \theta\).
   2. Hence show that a possible value of \(\tan \theta\) is \(2 \sqrt { 2 }\).

8

1. 1. Given that \(4 \sin x + 5 \cos x = 0\), find the value of \(\tan x\).
   2. Hence solve the equation \(( 1 - \tan x ) ( 4 \sin x + 5 \cos x ) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your values of \(x\) to the nearest degree.
2. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
   [0pt] [4 marks]

3. 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]

4. Solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$

1. (a) Write down the exact value of \(\cos \frac { \pi } { 6 }\).

The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(b) Use the trapezium rule with three equally-spaced ordinates to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(c) Using your answer to part (b), find an estimate for the area of \(S\).

5. (a) Given that $$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0 .$$ (b) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0 .$$

3. (a) Given that $$5 \cos \theta - 2 \sin \theta = 0 ,$$ show that \(\tan \theta = 2.5\) (b) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.

4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta ,$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).

5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

5

1. 1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
   2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
   (3 marks)

7

1. Solve the equation \(\sec x = - 5\), giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
2. Show that the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ can be written in the form $$\sec ^ { 2 } x = 25$$
3. Hence, or otherwise, solve the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
   (3 marks)

4

1. By using a suitable trigonometrical identity, solve the equation $$\tan ^ { 2 } \theta = 3 ( 3 - \sec \theta )$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
2. Hence solve the equation $$\tan ^ { 2 } \left( 4 x - 10 ^ { \circ } \right) = 3 \left[ 3 - \sec \left( 4 x - 10 ^ { \circ } \right) \right]$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 90 ^ { \circ }\).

6

1. Show that $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) }$$ can be written as \(\operatorname { cosec } ^ { 2 } x\).
2. Hence solve the equation $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) } = \operatorname { cosec } x + 3$$ giving the values of \(x\) to the nearest degree in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
3. Hence solve the equation $$\frac { \sec ^ { 2 } \left( 2 \theta - 60 ^ { \circ } \right) } { \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) + 1 \right) \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) - 1 \right) } = \operatorname { cosec } \left( 2 \theta - 60 ^ { \circ } \right) + 3$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).