C3 Trigonometry - Trigonometric equations - Physics & Maths Tutor
C3 Trigonometry - Trigonometric equations
1. (a) Express 5 cos x ? 3 sin x in the form R cos(x + ), where R > 0 and 0 < < 1 . 2
(4)
(b) Hence, or otherwise, solve the equation 5 cos x ? 3 sin x = 4
for 0 x < 2 ,giving your answers to 2 decimal places.
(5) (Total 9 marks)
? ? their 0.27), rather than applying the correct method of (2? ? their principal angle ? their ? ). Premature rounding caused a significant number of candidates to lose at least 1 accuracy mark, notably with a solution of 0.28c instead of 0.27c.
2. Solve for 0 x 180?.
cosec2 2x ? cot 2x = 1
3. (a) Use the identity cos2 + sin2 = 1 to prove that tan2 = sec2 ? 1.
(b) Solve, for 0 < 360?, the equation 2 tan2 + 4 sec + sec2 = 2
(Total 7 marks)
(2)
(6) (Total 8 marks)
Edexcel Internal Review
1
C3 Trigonometry - Trigonometric equations
4. (a) Use the identity cos(A + B) = cosA cosB ? sinA sinB, to show that
cos 2A = 1 ? 2sin2A (2)
The curves C1 and C2 have equations C1: y = 3sin 2x C2: y = 4 sin2x ? 2cos 2x
(b) Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation 4cos 2x + 3sin 2x = 2 (3)
(c) Express 4cos2x + 3sin 2x in the form R cos(2x ? ), where R > 0 and 0 < < 90?, giving the value of to 2 decimal places. (3)
(d) Hence find, for 0 x < 180?, all the solutions of 4cos 2x + 3sin 2x = 2
giving your answers to 1 decimal place.
(4) (Total 12 marks)
5. (a) Write down sin 2x in terms of sin x and cos x.
(b) Find, for 0 < x < , all the solutions of the equation cosec x ? 8 cos x = 0
giving your answers to 2 decimal places.
(1)
(5) (Total 6 marks)
Edexcel Internal Review
2
C3 Trigonometry - Trigonometric equations
6. (a) (i) By writing 3 = (2 + ), show that sin 3 = 3 sin ? 4 sin3.
(ii) Hence, or otherwise, for 0 < < , solve 3
8 sin3 ? 6 sin + 1 = 0. Give your answers in terms of .
(b) Using sin( - ) = sin cos - cos sin , or otherwise, show that sin15? = 1 ( 6 - 2). 4
(4)
(5)
(4) (Total 13 marks)
7. (a) Given that sin2 + cos2 1, show that 1 + cot2 cosec2.
(b) Solve, for 0 < 180?, the equation 2cot2 ? 9cosec = 3,
giving your answers to 1 decimal place.
(2)
(6) (Total 8 marks)
Edexcel Internal Review
3
C3 Trigonometry - Trigonometric equations
8. (a) Using sin2 + cos2 1, show that cosec2 ? cot2 1.
(b) Hence, or otherwise, prove that cosec4 ? cot4 cosec2 + cot2.
(c) Solve, for 90? < < 180?, cosec4 ? cot4 = 2 ? cot .
(2)
(2)
(6) (Total 10 marks)
9. (a) Show that
(i)
cos 2 x cos x - sin x,
cos x + sin x
x
(n
-
1 4
)
,
n
(ii)
1 2
(cos 2x - sin
2x) cos2
x - cos
x sin
x-
1 2
(b) Hence, or otherwise, show that the equation
cos
cos 2 cos + sin
=
1 2
can be written as
sin 2 = cos 2.
(c) Solve, for 0 2, sin 2 = cos 2,
giving your answers in terms of .
(2) (3)
(3)
(4) (Total 12 marks)
Edexcel Internal Review
4
C3 Trigonometry - Trigonometric equations
10. f(x) = 12 cos x ? 4 sin x. Given that f(x) = R cos(x + ), where R 0 and 0 90?, (a) find the value of R and the value of . (4)
(b) Hence solve the equation 12 cos x ? 4 sin x = 7
for 0 x 360?, giving your answers to one decimal place. (5)
(c) (i) Write down the minimum value of 12 cos x ? 4 sin x. (1)
(ii) Find, to 2 decimal places, the smallest positive value of x for which this minimum value occurs. (2) (Total 12 marks)
11. (a) Given that 2 sin( + 30)? = cos( + 60)?, find the exact value of tan ?.
(5)
(b) (i)
Using the identity cos (A + B) cos A cos B ? sin A sin B, prove that cos 2A 1 ? 2 sin2 A.
(ii) Hence solve, for 0 x < 2, cos 2x = sin x,
giving your answers in terms of .
(iii)
Show that sin 2y tan y + cos 2y 1, for 0 y <
1 2
.
(2)
(5) (3) (Total 15 marks)
Edexcel Internal Review
5
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