C3 Trigonometry - Trigonometric identities

C3 Trigonometry - Trigonometric identities



1. (a) Express 3 cos + 4 sin in the form R cos( ? ), where R and are constants, R > 0 and 0 < < 90?. (4)

(b) Hence find the maximum value of 3 cos + 4 sin and the smallest positive value of for which this maximum occurs. (3)

The temperature, f (t), of a warehouse is modelled using the equation f(t) = 10 + 3 cos(15t)? + 4 sin(15t)?,

where t is the time in hours from midday and 0 t < 24.

(c) Calculate the minimum temperature of the warehouse as given by this model. (2)

(d) Find the value of t when this minimum temperature occurs.

(3) (Total 12 marks)

2. (a) Use the double angle formulae and the identity

cos(A + B) cosA cosB ? sin A sinB

to obtain an expression for cos 3x in terms of powers of cos x only. (4)

(b) (i) Prove that

cos x + 1 + sin x 2sec x, x (2n + 1) .

1 + sin x cos x

2

(4)

Edexcel Internal Review

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C3 Trigonometry - Trigonometric identities



(ii) Hence find, for 0 < x < 2, all the solutions of cos x + 1 + sin x = 4 .

1 + sin x cos x

3.

(3) (Total 11 marks)

The diagram above shows an oscilloscope screen. The curve shown on the screen satisfies the equation

y = 3 cos x + sin x.

(a) Express the equation of the curve in the form y = Rsin(x + ), where R and are constants, R > 0 and 0 < < . 2 (4)

Edexcel Internal Review

2

C3 Trigonometry - Trigonometric identities



(b) Find the values of x, 0 x < 2, for which y = 1.

(4) (Total 8 marks)

4. (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)

5. (a) Given that cos A = 3 , where 270? < A < 360?, find the exact value of sin 2A. 4

(5)

(b) (i) Show that cos 2x + + cos 2x - cos 2x 3 3

(3)

Given that y = 3sin 2 x + cos 2x + + cos 2x - , 3 3

(ii) show that dy = sin 2x dx

(4) (Total 12 marks)

Edexcel Internal Review

3

C3 Trigonometry - Trigonometric identities



6. (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 .

7. (a) Differentiate with respect to x (i) x2e3x+2,

(ii) cos(2x3 ) . 3x

(2) (3)

(3)

(4) (Total 12 marks)

(4) (4)

Edexcel Internal Review

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C3 Trigonometry - Trigonometric identities



(b) Given that x = 4 sin(2y + 6), find dy in terms of x. dx

(5) (Total 13 marks)

8. 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)

9. (a) Given that 2 sin( + 30)? = cos( + 60)?, find the exact value of tan ?.

(5)

Edexcel Internal Review

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