Download Trigonometry 2

Trigonometry 2 – Core 2 Revision
1.
Solve the equation 4 + 3 sin (2x – 1) = 6 for 0  x 
equation for 0  x 2 

. Write down one further solution to the
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
3  sin 2   2  cos 
2  cos 
(2)
(b)
Use the identity from part (a) to show that the equation
3  sin 2 2 x  5
2  cos 2 x 4
can be written in the form cos 2x = 3 .
4
(1)
South Wolds Comprehensive School
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(c)
Solve the equation
cos 2x = 3
4
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)
South Wolds Comprehensive School
2
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

;
4
(ii)
cos

;
6
(iii)
tan

.
3
(3)
The diagram shows the graphs of
y = sin2 x and y =
1
for 0  x  .
2
y
1
2

O
(b)
Solve sin2 x =
x
1
for 0  x  
2
(3)
(c)
Hence solve sin2 x 
1
for 0  x  
2
(2)
(d)
Prove that
sin2 x 
1
1
 cos2 x <
2
2
(2)
(Total 10 marks)
South Wolds Comprehensive School
3
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 in part (b). Hence explain why the only possible value
1
of sin  which will satisfy it is .
2
(3)
(ii)
Write down the values of  for which sin  =
1
and 0    2.
2
(2)
(iii)
For the smaller of these values of 0, write down the exact values, in surd form,
of tan  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


cos x    0.5,
6

in the interval 0 < x < 2 p, leaving your answers in terms of .
(6)
(Total 6 marks)
South Wolds Comprehensive School
4
13.
The angle x, measured in radians, satisfies the equation
2 sin2 x = 1 + cos x.
(a)
Verify that one root of this equation is  .
3
(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 sin2 x = 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  x    = 0.3
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)
South Wolds Comprehensive School
5
17.
The diagram shows the graphs of
y = cos2 x and y = sin x for 0  x  :
The graphs intersect each other at two points P and Q.
y
1
P
Q

O
(a)
x
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 School
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