Trigonometry 2 – Core 2 Revision

1.Solve the equation 4 + 3 sin (2x – 1) = 6 for 0 x. Write down one further solution to the equation for 0 x 2 

(Total 5 marks)

2.Given that

3sin + cos  = 0,

(a)find the value of tan ,

(2)

(b)find the values of  in the interval 0º  < 360º.

(2)

(Total 4 marks)

3.Solve the equation

5 tan (3x + 30°) = 2,

in the interval 0° x  180°, giving your answers correct to the nearest degree.

(6)

(Total 6 marks)

4.A graph has equation

y = cos 2x,

where x is a real number

(a)Draw a sketch of that part of the graph for which 0  x 2

(2)

(b)On your sketch show two of the line of symmetry which the complete graph possesses

(2)

(Total 4 marks)

5.(a)Prove the identity

(2)

(b)Use the identity from part (a) to show that the equation

can be written in the form cos 2x = .

(1)

(c)Solve the equation

cos 2x =

in the interval 0° x 180°, giving your answers to the nearest 0.1°.

(No credit will be given for simply reading values from a graph.)

(4)

(Total 7 marks)

6.(a)Given that

2cos2 –sin  = 1,

show that

2sin2 + sin  – 1 = 0.

(2)

(b)In this part of the question, no credit will be given for an approximate numerical method.

Hence find all the values of  in the interval 0 <  < 2 for which

2cos2 – sin  = 1,

giving each answer in terms of .

(4)

(c)Write down all the values of x in the interval 0 < x  for which

2cos2 2x – sin 2x = 1.

(2)

(Total 8 marks)

7.(a)Describe the geometrical transformation that maps the curve with equation y = sinx onto the curve with equation:

(i)y = 2 sinx;

(2)

(ii)y = –sinx;

(2)

(iii)y = sin(x – 30°).

(2)

(b)Solve the equation sin(θ – 30°) = 0.7, giving your answers to the nearest 0.1° in the interval 0° ≤ θ ≤ 360°.

(3)

(c)Prove that (cosx + sinx)2 + (cosx – sinx)2 = 2.

(4)

(Total 13 marks)

8.Solve the equation

cos (4x + 40) = 0.5

giving all solutions in the interval 0x < 180.

(No credit will be given for simply reading values from a graph.)

(Total 6 marks)

9.(a)Write down the exact values of:

(i)sin;

(ii)cos;

(iii)tan.

(3)

The diagram shows the graphs of

y = sin2x and y = for 0 x.

(b)Solve sin2x = for 0 x

(3)

(c)Hence solve sin2x for 0 x

(2)

(d)Prove that

sin2x cos2x

(2)

(Total 10 marks)

10.The angle  radians, where 0  2, satisfies the equation

3 tan  = 2 cos .

(a)Show that

3 sin  = 2 cos2.

(1)

(b)Hence use an appropriate identity to show that

2 sin2+ 3 sin  – 2 = 0.

(3)

(c)(i)Solve the quadratic equation inpart (b). Hence explain why the only possible value of sin  which will satisfy it is.

(3)

(ii)Write down the values of for which sin  = and 0  2.

(2)

(iii)For the smaller of these values of 0, write down the exact values, in surd form, oftan  and cos .

(2)

(iv)Verify that these exact values satisfy the original equation.

(1)

(Total 12 marks)

11.Solve the equation

sin(2x + 20°) = 0.5

giving all solutions in the interval 0° < x < 360°

No credit will be given for simply reading the values from the graph.

(Total 6 marks)

12.Solve the equation

in the interval 0 < x < 2 p, leaving your answers in terms of .

(6)

(Total 6 marks)

13.The angle x, measured in radians, satisfies the equation

2 sin2x = 1 + cos x.

(a)Verify that one root of this equation is .

(2)

(b)Use a trigonometric identity to show that

2 cos2x +cos x – 1 = 0.

(2)

(c)Hence find all the roots of the equation

2 sin2x = 1 + cos x

in the interval 0 x < 2.

(4)

(Total 8 marks)

14.It is given that x satisfies the equation

2 cos2x = 2 + sin x.

(a)Use an appropriate trigonometrical identity to show that

2 sin2x + sin x = 0.

(2)

(b)Solve this quadratic equation and hence find all the possible values of x in the interval
0 x < 2.

(6)

(Total 8 marks)

15.Find, in radians, the values of x in the interval 0 x 2 for which

sin = 0.3

Give your answers to 3 significant figures.

(Total 6 marks)

16.(a)Given that 3 cos 5x = 4 sin 5x, write down the value of tan 5x.

(1)

(b)Hence, find all solutions of the equation

3 cos 5x = 4 sin 5x

in the interval 0° x 90°, giving your answers correct to the nearest 0.1°.

(4)

(Total 5 marks)

17.The diagram shows the graphs of

y = cos2x and y = sin x for 0 x:

The graphs intersect each other at two points P and Q.

(a)Use a trigonometric identity to show that the x-coordinates of P and Q satisfy the equation

sin2x + sin x – 1 = 0.

(2)

(b)(i)Solve this quadratic equation.

(2)

(ii)Show that the only possible value for sin x is approximately 0.618.

(2)

(c)Find the x-coordinates of P and Q, giving each answer to two decimal places.

(3)

(Total 9 marks)

South Wolds Comprehensive School1