Double and Compound Angle Revision – Core 4
1.Find the general solution, in radians, of the equation
sin 2x + cos x = 0
(Total 5 marks)
2.(a)Use the identity sin(A + B) + sin(A – B)= 2 sin A cos B to prove that:
(i)sin 2 = 2 sin cos ;
(1)
(ii)sin3 +sin = 4 sin – 4 sin3 .
(4)
(b)Show that (2s – 1) is a factor of 8s3 – 8s + 3.
(1)
(c)Solve the equation
sin 3 +sin =
giving your answers in radians in the interval 0 < .
(9)
(Total 15 marks)
3.Solve the equation
3 sin sin2 = 2cos3,
giving all solutions in radians in the interval 0 < < 2, leaving your answers in terms of .
No credit will be given for a numerical approximation or for a numerical answer from a calculator without supporting working.
(Total 6 marks)
4.Solve the equation
3 cos 2– cos + 1 = 0
giving all solutions in degrees to the nearest degree in the interval 0° 360°.
(Total 6 marks)
5.Solve the equation
8 tan = tan 2,
giving all solutions to the nearest 0.1° in the interval –90° < < 90°.
(Total 6 marks)
6.Determine all the values of x between 0 and 2 which satisfy the equation
5 cos 2x + 3 sin x= 4,
giving your answers in radians to three significant figures or exactly in terms of .
.
(Total 7 marks)
7.(a)Solve the equation sec x = 2 for 0 x 2.
(2)
(b)Use the identity cos (A + B) cos A cos B – sin A sin B to show that
cos 2x = 2 cos2x – 1.
(2)
(c)Hence solve the equation
cos 2x + 3 cos x – 1 = 0 for 0 x 2.
(5)
(Total 9 marks)
8.(a)Given that sin = , where is an obtuse angle, find the exact value of cos .
(2)
(b)Given also that cos = , where is an acute angle, find the exact value of sin( + ).
(3)
(Total 5 marks)
9.(a)Show that tan(45° + ) =.
(2)
(b)Hence obtain the exact value of tan 105° in the form a + b, where a and b are integers to be found.
(4)
(Total 6 marks)
10.(a)Use the expansion of cos (A + B) to show that
cos 3x = 4 cos3x – 3 cos x.
(3)
(b)Hence find all solutions in the interval 0 < x of the equation
2 cos 3x = cos x,
giving each answer in radians to two decimal places.
(6)
(Total 9 marks)
11.(a)By factorising cos4x – sin4x, or otherwise, show that
cos4x – sin4x = cos 2x.
(3)
(b)Hence, or otherwise, find the range of values of x in the interval 0 xfor which
cos4x – sin4x < 0.
(3)
(Total 6 marks)
12.(a)Express 2cos x – sin x in the form R cos(x + ),where R is a positive constant and is an angle between 0º and 360º.
(2)
(b)Given that 0º x < 360º,
(i)solve2cos x – sin x = l,
(3)
(ii)deduce the solution set of the inequality
2cos x – sin x 1.
(2)
(Total 7 marks)
13.(a)Express the function sin °+ 2 cos in the form R sin ( + ), stating the values of R and to 3 significant figures.
(2)
(b)
The diagram shows a rectangle ABCD. The mid point of AB is M. The length MC is a and the size of angle BMC is.
(i)Express the perimeter of the rectangle in terms of a and. Hence state, in terms of a, the maximum possible value of the perimeter as varies.
(3)
(ii)Find, in terms of a, the maximum possible value of the area of the rectangle as
varies.
(3)
(Total 8 marks)
14.(a)Prove the identity
(4)
(b)Given that 0 < x, solve the equation
tan x + cot x = 22.
(3)
(Total 7 marks)
15.(a)Given that tan 1, show that
.
(3)
(b)By expressing cos x + sin x in the form R sin (x + a), solve, for 0° x 360°,
.
(5)
(Total 8 marks)
South Wolds Comprehensive School1