### Question

7  A curve is defined by the parametric equations
x = 3t − 2 sin t,            y = 5t + 4 cos t,

where $$0\leqslant t\leqslant 2\pi$$. At each of the points P and Q on the curve, the gradient of the curve is $$\frac{5}{2}$$.

(a) Show that the values of t at P and Q satisfy the equation 10 cos t − 8 sin t = 5.                                                                                        

(b) Express 10 cost − 8 sin t in the form $$R\cos (t+\alpha ), where \ R> 0> and \ 0< \alpha < \frac{1}{2}\pi$$. Give the exact
value of R and the value of $$\alpha$$  correct to 3 significant figures.                                                                                                      

(c) Hence find the values of t at the points P and Q.                                                                                                                                              

Ans

7 (a) Carry out division at least as far as  3x2+ kx

Obtain quotient 3x2 – 4 – 4

Confirm remainder is 9 AG

7 (b) Integrate to obtain at least  k1 x3 and k2 ln(3x + 2) terms

Obtain x3 – 2x2 – 4x + 3 ln(3x + 2)
(FT from quotient in part (a))

Apply limits correctly
Apply appropriate logarithm properties correctly
Obtain 125 ln64

7 (c) State or imply $$9x^{3}-6x^{2}-20x-8=(3x+2)(3x^{2}-4x-4)$$

(FT from quotient in part (a))
Attempt to solve cubic eqn to find positive value of x (or of 3ey )
Use logarithms to solve equation of form 3ey = k where k > 0 M1
Obtain $$\frac{1}{3}ln2$$ or exact equivalent

### Question

It is given that 3 sin 2θ = cos θ where θ  is an angle such that 0o < θ < 90o.
(a) Find the exact value of sin θ.                                                                                   

(b) Find the exact value of sec θ .                                                                                  

(c) Find the exact value of cos 2θ .                                                                                

Ans

(a) Express left-hand side in terms of sinθ and cosθ

Obtain 2 cos θ  – 2 sin θ
Attempt to express a cos θ + b sin θ in Rcos (θ+β) form

Confirm $$R=\sqrt{8}AG$$

Carry out necessary trigonometry and confirm $$\frac{1}{4}\pi AG$$

(b) Carry out correct process to find θ from $$\cos \left ( \theta +\frac{1}{4}\pi \right )=\frac{1}{\sqrt{8}}$$

Obtain 0.424

(c) Express integrand as $$\sqrt{8}\cos \left ( \frac{1}{2}x+\frac{1}{4}\pi \right )\ or\ as \ 2\cos \frac{1}{2}x-2\sin \frac{1}{2}x$$

Integrate to obtain $$k\sin (\frac{1}{2}x+\frac{1}{4}\pi )\ or \ 4\sin \frac{1}{2}x+4\cos \frac{1}{2}x$$

Obtain correct $$2\sqrt{8}\sin \left ( \frac{1}{2}x+\frac{1}{4}\pi \right )\ or 4\sin \frac{1}{2}x+4\cos \frac{1}{2}x$$

### Question

Solve the equation $$2 sin ( \theta + 30^o)+5 cos \theta = 2 sin \theta$$ for $$0^o<\theta<90^o$$.

Ans:

Express first term as $$2 sin \theta cos 30+2cos \theta sin 30$$
Divide by $$cos\theta$$ to produce linear equation in $$tan \theta$$
Obtain $$tan \theta = \frac{6}{2\sqrt{3}}$$ or 22.39…
Obtain 87.4

### Question

(a) Express $$5\sqrt{3}cos x + 5sin x$$ in the form $$R cos(x- \alpha )$$, where R>0 and $$0<\alpha<\frac{1}{2}\pi$$.
(b) As x varies, find the least possible value of
$$4+5\sqrt{3}cos x+5sinx$$,
and determine the corresponding value of x where $$-\pi<x<\pi$$.
(c) Find $$\int \frac{1}{(5\sqrt{3}cos3\theta+5sin3\theta)^2}d\theta$$

Ans:

1. State R=10
Use appropriate trigonometry to find $$\alpha$$
Obtain $$\alpha=\frac{1}{6}\pi$$
2. State -6
Attempt to find x from their $$cos(x-\alpha)=-1$$
Obtain $$x-\frac{1}{6}\pi=-\pi$$ and hence $$-\frac{5}{6}\pi$$
3. State integrand of form $$k_1sec^2(3\theta-\frac{1}{6}\pi)$$
Integrate to obtain form $$k_2tan(3\theta-\frac{1}{6}\pi)$$
Obtain $$\frac{1}{300}tan(3\theta-\frac{1}{6}\pi)+c$$

### Question

Solve the equation $$sec^2\theta cot\theta =8$$ for $$0<\theta<\pi$$.

Ans:

State $$\frac{1}{cos^2\theta} \times \frac{cos \theta}{sin \theta} =8$$
Attempt use of $$sin 2\theta$$ identity to obtain $$sin 2\theta = k$$
Obtain $$sin 2\theta = \frac{1}{4}$$
Use correct process to find two values of $$\theta$$ between 0 and $$\pi$$.
Obtain 0.126 and 1.44

Alternative method for question 2

State $$\frac{1+tan^2 \theta}{tan \theta} =8$$
Attempt solution of 3-term quadratic equation to find values of $$tan\theta$$
Obtain $$tan \theta = \frac{8± \sqrt{60}}{2}$$
Solve $$tan \theta = ….$$ to find two values of $$\theta$$ between 0 and $$\pi$$.
Obtain 0.126 and 1.44

### Question

A curve is defined by the parametric equations
x = 3t − 2 sin t,                    y = 5t + 4 cos t,
where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, the gradient of the curve is $$\frac{5}{2}$$

(a) Show that the values of t at P and Q satisfy the equation 10 cost − 8 sin t = 5.                                                                                       

(b) Express 10 cost − 8 sin t in the form $$R \cos(t+\alpha ), \ where \ R> 0\ and \ 0< \alpha \frac{1}{2}\pi$$. Give the exact
value of R and the value of $$\alpha$$ correct to 3 significant figures.                                                                                                    

(c) Hence find the values of t at the points P and Q.                                                                                                                                            

Ans

(a) Obtain $$\frac{dx}{dt}=3-2\cos t \ and \ \frac{dy}{dt}=5-4\sin t$$

Equate expression for $$\frac{dy}{dx}\ to \ \frac{5}{2}$$

Obtain $$10\cos t-8\sin t=5$$

7(b) State $$R=\sqrt{164}$$ or exact equivalent

Use appropriate trigonometry to find $$\alpha$$

Obtain 0.675 with no errors seen

(c) Carry out correct method to find one value of t
Obtain 0.495
Carry out correct method to find second value of t
Obtain 4.44

### Question

It is given that 3 sin 2 θ = cos θ  where θ is an angle such that 0o < 1 < 90o.
(a) Find the exact value of sin θ .                                                                                          

(b) Find the exact value of sec θ.                                                                                          

(c) Find the exact value of cos 2 θ .                                                                                      

Ans

(a) Use sin 2 θ = 2sin θ cos  θ

Obtain $$\sin \theta =\frac{1}{6}$$

(b) Use correct identity or identities to find value of secθ

Obtain $$\frac{6}{\sqrt{35}}$$ or exact equivalent

(c) Use correct identity or identities to find value of cos2θ

Obtain $$\frac{17}{18}$$ or exact equivalent

### Question

(a) Show that (sec x + cos x)2 can be expressed as sec2x + a + b cos 2x,                     where  a and b are constants
to be determined.                                                                                               

Ans

3(a) Expand to obtain integrand of form sec2 x + k1 + k2 cos 2x

Obtain correct sec2  $$x+\frac{5}{2}+\frac{1}{2}\cos 2x$$

3(b) Integrate to obtain at least terms of form k3 tan x and k4 sin 2x

Obtain correct tan $$x+\frac{5}{2}x+\frac{1}{4}2x$$

Apply limits correctly to integral involving at least two terms

Obtain $$\frac{5}{4}+\frac{5}{8}\pi \ or \ exact equivalent$$

### Question

By first expanding sin(θ + 30), solve the equation sin(θ + 30) cosec θ = 2 for 0o < θ < 360.    

Ans

2  Express sin ( θ + 30 ) as sin θ cos 30 + cos θ sin 30

Use  cosecθ=$$\frac{1}{\sin \theta }$$

Correctly obtain a linear equation in tanθ or cotθ

Obtain tanθ = $$\frac{1}{4-\sqrt{3}}, \frac{4+\sqrt{3}}{13}, \frac{1} {2.26795} \ or \ 0.440…$$

Obtain 23.8
Obtain 203.8

### Question

(a)By first expanding cos (2θ + θ), show that cos 3θ = 4 cos3θ – 3 cos θ.

(b)Find the exact value of 2 cos3  $$\left ( \frac{5}{18}\pi \right ) – \frac{3}{2} cos \left ( \frac{5}{18} \pi \right )$$ .

(c)Find $$\int (12cos^{3}x – 4 cos^{3} 3x)dx$$ .

(a)State cos 2θ cos θ – sin 2 θ sin θ

Attempt correct relevant identities to express in terms of cos θ only

Confirm 4 cos3 θ -3 cos  θ with sufficient detail

(b)Use identity with θ = $$\frac{5}{18}\pi$$

Obtain $$\frac{1}{2} cos \frac{5}{6}\pi and hence – \frac{1}{4}\sqrt{3}$$

(c)Express integrand in form k1 (cos 3x + 3 cos x) + k2 (cos 9x + 3 cos 3x)

Obtain correct integrand 9 cos x –  cos9 x

Integrate to obtain form k3 sin x + k4  sin 9x

Obtain correct 9 sin x – $$\frac{1}{9}$$  sin 9x

### Question

The polynomial p (x) is defined by

p(x) = 4x3 + 16x2 + 9x -15.

(a)Find the quotient when p(x) is divided by (2x+3), and show that the remainder is −6.

(b) Find $$\int \frac{p(x)}{2x+3}dx$$ .

(c)Factorise p (x) +6 completely and hence solve the equation

p (cosec 2θ) + 6 = 0

for 00 < θ < 1350.

Ans:

(a)Carry out division at least as far as 2x2 + kx

Obtain quotient 2x2 + 5x – 3

Confirm remainder is -6.

(b)Integrate to obtain at least k1 x 3 and  k2 ln (2x 3) + terms

Obtain $$\frac{2}{3}x^{3} + \frac{5}{2}x^{2} – 3x – 3 In (2x+3)$$

(c)State or imply p(x) + 6 = (2x + 3) (2x2 + 5x -3)

Conclude (2x + 3) (2x – 1) (x + 3)

State or imply  sin 2θ = $$-\frac{2}{3}$$ or sin 2θ = $$-\frac{1}{3}$$ or both

Carry out correct process to find θ in at least one case

Obtain 99.7 and 110.9

### Question

(a) Showing all necessary working, solve the equation $$\sec \Theta \csc \Theta =7$$  for $$0^{\circ}< \Theta < 90^{\circ}$$.

(b) Showing all necessary working, solve the equation

$$\sin (\beta +20^{\circ})+\sin (\beta -20^{\circ})=6\cos \beta$$  for $$0^{\circ}< \Theta < 90^{\circ}$$.

6(a) Express equation as $$\frac{1}{cos\alpha+sin\alpha }=7$$

Attempt use of identity for sin 2α or attempt to obtain a quadratic equation in terms of any one of the following: $$sin^{2}\alpha,cos^{2}\alpha ,cot^{2}\alpha or tan^{2}\alpha$$

Obtain  sin 2α =$$\frac{2}{7}$$or a correct 3 term quadratic equation, equated to zero in any one of the following: $$sin^{2}\alpha,cos^{2}\alpha ,cot^{2}\alpha or tan^{2}\alpha$$

Attempt correct process to find at least one correct value of α  Obtain 8.3 and 81.7 and no others between 0 and 90

6(b) Simplify left-hand side to obtain 2sin cos20 β °
Attempt to form equation where tan β is only variable, tan β≠3

Obtain  $$tan\beta =\frac{3}{cos20\AA }$$

Obtain β= 72.6 and no others between 0 and 90

### Question

(i) Show that $$2 cosec 21 cot\Theta =cosec^{2}$$

(ii) Hence show that cosec^{2}15Å tan 15Å = 4.

(iii) Solve the equation $$2cosec\Theta cot\frac{\Theta }{2}+cosec\frac{\Theta }{2}=12$$ for $$−360Å <\Theta < 360Å$$ Show all necessary
working.

(i)State or  imply $$\csc 2\theta =\frac{1}{2\sin \theta \cos \theta }$$

Attempt to express left-hand side in terms of  $$\sin \theta$$ and $$\cos \theta$$ only

Simplify to confirm $$\csc ^{2}\theta$$

(ii) Use identity to express left-hand side in terms of $$\sin 30$$ or $$\csc 30$$

Obtain$$\frac{2}{\sin 30}$$ or $$2\csc 30$$ and confirm 4

(iii) Solve quadratic equation of the form$$k\csc ^{2}\frac{\phi }{2}+\csc \frac{\phi }{2}-12=0$$ or $$12\sin ^{2}\frac{\phi }{2}-\sin \frac{\phi }{2}-k=0$$ correctly for\csc \frac{1}{2}\phi  \)or $$\sin \frac{1}{2}\phi$$ to find two values of $$\csc \frac{1}{2}\phi$$\) or $$\sin \frac{1}{2}\phi$$

Obtain $$\sin \frac{1}{2}\phi =-\frac{1}{4},\frac{1}{3}$$

Use correct process to find at least one correct value of $$\phi$$ from $$\sin \frac{1}{2}\phi =\pm \frac{1}{4},\pm \frac{1}{3}$$

Obtain any two of -331.0,-29.0,38.9,321.1.

### Question The diagram shows the curve with equation y = sin 2x + 3 cos 2x for 0 ≤ x ≤ 0. At the points P and Q on the curve, the gradient of the curve is 3.

(i) Find an expression for $$\frac{dy}{dx}$$

(ii) By first expressing $$\frac{dy}{dx}$$  in the form R cos (2x + \alpha), where $$R>0 and 0<\alpha <\frac{1}{2}\pi$$  R , find the x-coordinates of P and Q, giving your answers correct to 4 significant figures. 

7(i) State expression of form $$k_{1}cos2x+k_{2}sin2x$$

7(ii) State$$R=\sqrt{40} or 6.324…$$

Use appropriate trigonometry to find α

Obtain 1.249…

Equate their $$Rcos(2x+\alpha ) to 3 and find cos^{-1}(3\div R)$$

Carry out correct process to find one value of α

Obtain 1.979

Carry out correct process to find second value of α within the range

Obtain 3.055

### Question

Solve the equation $$sec^{2}\Theta$$= 3 cosec 1 for 0Å < 1 < 180Å.

State$$\frac{1}{cos^{2}\theta }=\frac{3}{sin\Theta }$$ or $$1+tan^{2}\Theta =\frac{3}{sin\Theta }$$

Solve 3-term quadratic equation to find value between –1 and 1 for sinθ

Obtain  $$sin\Theta =\frac{1}{6}(-1+\sqrt{37})$$ and hence 57.9
Obtain 122.1 and no others between 0 and 180

### Question

(a) Showing all necessary working, solve the equation $$sec\alpha cosec\alpha = 7$$  for $$0^{\circ} < \alpha < 90^{\circ}$$

(b) Showing all necessary working, solve the equation

$$sin(\beta +20)+ sin(\beta – 20) = 6 cos\beta$$      for    $$0^{\circ}< \beta< 90^{\circ}$$

(i) Substitute x=−2 and equate to zero

Obtain
or equivalent and hence $$a=7$$

Attempt either division by $$x+2$$ and reach partial quotient $$x^{2}+kx$$  where  k is numeric or use of identity or inspection or synthetic  division

Obtain quotient $$x^{2}+5x+4$$

Conclude with  $$(x+1)(x+2)(x+4)$$ ### Question

(a) Showing all necessary working, solve the equation $$sec\alpha cosec\alpha = 7$$  for $$0^{\circ} < \alpha < 90^{\circ}$$

(b) Showing all necessary working, solve the equation

$$sin(\beta +20)+ sin(\beta – 20) = 6 cos\beta$$      for    $$0^{\circ}< \beta< 90^{\circ}$$

(i) Substitute x=−2 and equate to zero

Obtain
or equivalent and hence $$a=7$$

Attempt either division by $$x+2$$ and reach partial quotient $$x^{2}+kx$$  where  k is numeric or use of identity or inspection or synthetic  division

Obtain quotient $$x^{2}+5x+4$$

Conclude with  $$(x+1)(x+2)(x+4)$$ ### Question

The parametric equations of a curve are

x = 2t − sin 2t, y = 5t + cos 2t, for$$0 ≤ t \frac{1}{2}\pi$$ .

At the point P on the curve, the gradient of the curve is 2.

(i) Show that the value of the parameter at P satisfies the equation 2 sin 2t − 4 cos 2t = 1.

(ii) By first expressing 2 sin 2t − 4 cos 2t in the form$$R sin2t − \Theta$$, where$$R > 0 and 0 < ! <\Theta$$

(i) Obtain expression for $$\frac{dy}{dx}$$ with numerator quadratic, denominator linear

Obtain$$\frac{3t^{2}-6t}{2t+4}$$

Identify t = 3 at P
Obtain $$\frac{9}{10}$$or equivalent

(ii) Equate first derivative to zero and obtain non-zero value of t
Obtain t = 2
Substitute to obtain (12,- 4)

(iii) Equate expression for gradient to m and rearrange to confirm$$3t^{2}-(2m+6)-4m$$=0

Attempt solution of quadratic inequality or equation resulting from
discriminant

Obtain critical values -$$\sqrt{72}-9,m\geqslant \sqrt{72-9}$$

Conclude
$$m\leqslant \sqrt{72-9},m\geqslant \sqrt{72}-9$$

### Question

Solve the equation $$sec^{2}\Theta +tan^{2}\Theta =5tan\Theta 4 for 0Å < 1 < 180Å. Show all necessary working.  Use identity \(sec^{2}\Theta =1+tan^{2}\Theta$$

Attempt solution of quadratic equation to find two values of tanθ

Obtain $$tan\Theta =-\frac{1}{2},3$$

Obtain 71.6 and 153.4 and no others between 0 and 180

### Question

(i) Given that$$tan2\Theta cot\Theta = 8$$, show that $$tan^{2}\Theta =\frac{3}{4}$$

(ii) Hence solve the equation tan 21 cot 1 = 8 for 0Å < 1 < 180Å

.

(i) Use identity $$cot\Theta =\frac{1}{tan\Theta }$$

Attempt use of identity for $$tan2\Theta$$

Confirm given $$tan^{2}\Theta =\frac{3}{4}$$

(ii) Obtain 40.9 B1
Obtain 139.1

### Question

Solve the equation $$5 tan 2\Theta = 4cot\Theta for 0Å < 1 < 180Å.$$

Use cot\Theta =1\div tan\Theta

Form equation involving tanθ only and with no denominators involving θ

Obtain$$tan^{2}\Theta =\frac{2}{7}$$

Obtain 28.1
Obtain 151.9
Allow other valid methods

### Question

(i) Show that sin 2x cot x Ξ 2 cos2x. 

(ii) Using the identity in part (i),

(a) find the least possible value of
3 sin 2x cot x + 5 cos 2x + 8
as x varies, 

(b) find the exact value of $$\int_{\frac{1}{8}\pi }^{\frac{1}{6}\pi}$$ cosec 4x tan 2x dx. 

Ans:

8 (i) State cos 2sin x cos x . $$\frac{cos x}{sin x}$$
Simplify to confirm 2cos2x

(ii) (a) Use cos2x= 2cos2 x-1
Express in terms of cos x
Obtain 16cos2 x + 3 or equivalent
State 3, following their expression of form a cos2 x + b

(b) Obtain integrand as $$\frac{1}{2}sec^{2}2x$$
Integrate to obtain form k tan 2x
Obtain correct $$\frac{1}{2}tan2x$$
Apply limits correctly dep
Obtain $$\frac{1}{4}\sqrt{3}-\frac{1}{4}$$ or exact equivalent

### Question

(i) Express 8 sin θ + 15 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places. 

(ii) Hence solve the equation

8 sin θ + 15 cos θ = 6

for 0° ≤ 1 ≤ 360°. 

Ans:

3 (i) State or imply R = 17
Use appropriate formula to find α
Obtain 61.93 A1

(ii) Attempt to find at least one value of θ + α
Obtain one correct value of θ (97.4 or 318.7)
Carry out correct method to find second answer
Obtain second correct value and no others between 0 and 360

### Question

(i) Prove that 2 cosec 2θ tan θ ≡ sec2 θ. 

(ii) Hence

(a) solve the equation 2 cosec 2θ tan θ = 5 for 0 < θ < π, 

(b) find the exact value of  $$\int_{0}^{\frac{1}{6}\pi }$$ 2 cosec 4x tan 2x dx. 

Ans:

6 (i) State or imply $$cosec2\theta=\frac{1}{sin2\theta }$$
Express left-hand side in terms of sinθ and cosθ
Obtain given answer sec2 θ correctly

(ii) (a) State or imply $$cos\theta =\frac{1}{\sqrt{5}}$$ or tan θ = 2 at least
Obtain 1.11 or awrt 1.11, allow 0.353π
Obtain 2.03 or awrt 2.03 , allow 0.648π and no other values between 0 and π

(b) State integrand as sec2 2x
Integrate to obtain expression of form k tan mx
Obtain correct $$\frac{1}{2}$$ tan 2x
Obtain $$\frac{1}{2}\sqrt{3}$$ or exact equivalent

Question

(i) Express $$3\cos \Theta +\sin \Theta$$   in the form $$R\cos (\Theta -\alpha )$$, where R > 0 and $$0^{\circ}< \alpha < 90^{\circ}$$, giving the exact value of R and the value of correct to 2 decimal places.
(ii) Hence solve the equation

3 cos 2x + sin 2x = 2,

giving all solutions in the interval $$0^{\circ}\leq x\leq 360^{\circ}$$.

(i) State $$R=\sqrt{10}$$
Use trig formula to find α
Obtain α = 18.43 with no errors seen
(ii) Carry out evaluation of $$\cos ^{-1}\left ( \frac{2}{R} \right )\approx 50.77^{\circ}$$
Carry out correct method for one correct answer
Obtain one correct answer e.g. 34.6°
Carry out correct method for a further answer
Obtain remaining 3 answers 163.8°, 214.6°, 343.8° and no others in the range

Question

(i) Express 5 sin 2θ + 2 cos 2θ in the form Rsin(2θ + α), where R > 0 and 0◦ < α < 90◦, giving the exact value of R and the value of α correct to 2 decimal places.
Hence
(ii) solve the equation

5 sin 2θ + 2 cos 2θ = 4,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦

(iii) determine the least value of $$\frac{1}{(10\sin 2\Theta +4\cos 2\Theta )^{2}}$$ as θ varies.

(i) State $$R=\sqrt{29}$$
Use trig formula to find α
Obtain $$\alpha =21.80^{\circ}$$ with no errors seen

(ii) Carry out evaluation of $$\sin ^{-1}(\frac{4}{R})\approx 49.97^{\circ}$$
Carry out correct method for one correct answer
Obtain one correct answer e.g. $$13.1^{\circ}$$

Carry out correct method for a further answer
Obtain remaining 3 answers $$55.1^{\circ}$$,$$193.1^{\circ}$$,$$235.1^{\circ}$$and no others in the range
(iii) Greatest value of $$10\sin 2\Theta +4\cos 2\Theta =2\sqrt{29}$$

$$\frac{1}{116}$$

Question

(i) Show that $$12\sin ^{2}x\cos ^{2}x=\frac{3}{2}(1-\cos 4x)$$
(ii) Hence show that

$$\int _{\frac{1}{4}\pi }^{\frac{1}{3}\pi }12\sin ^{2}x\cos ^{2}xdx=\frac{\pi }{8}+\frac{3\sqrt{3}}{16}$$.

(i) Either
Use $$\sin 2x=2\sin x\cos x$$ to convert integrand to $$k\sin ^{2}2x$$

Use $$\cos 4x=1-2\sin ^{2}2x$$

State correct expression $$\frac{1}{2}-\frac{1}{2}\cos 4x$$ or equivalent.

Or
Use $$\cos ^{2}x=\frac{1-\cos 2x}{2}$$ and/or $$x=\frac{1-\cos 2x}{2}$$  to obtain an equation in cos 2x only

Use $$\cos ^{2}2x=\frac{1+\cos 4x}{2}$$.

State correct expression $$\frac{1}{2}-\frac{1}{2}\cos 4x$$ or equivalent.

(ii) State correct integral $$\frac{3}{2}x-\frac{3}{8}\sin 4x$$
Attempt to substitute limits, using exact values

Question

Solve the equation $$2\cos 2\Theta =4\cos \Theta -3$$,for $$0^{\circ}\leq \Theta \leq 180^{\circ}$$.

Make relevant use of the cos 2θ formula
Obtain a correct quadratic in cos θ
Solve a quadratic in cos θ
Obtain answer θ = 60 and no others in the range
(Ignore answers outside the given range)

Question

(i) Show that $$(2\sin x+\cos x)^{2}$$ can be written in the form $$\frac{5}{2}+2\sin 2x-\frac{3}{2}\cos 2x$$.
(ii) Hence find the exact value of  $$\int ^{\frac{1}{2}\pi }_{0}(2\sin x+\cos x)^{2}dx.$$

(i)Expand  to obtain $$4\sin ^{2}x+4\sin x\cos x+\cos ^{2}x$$.

Use $$2\sin x\cos x=\sin 2x$$.

Attempt to express $$\sin ^{2}x$$ or $$\cos ^{2}x$$(or both) in terms of $$\cos 2x$$

Obtain correct $$\frac{1}{2}k(1-\cos 2x)$$ for their or equivalent $$k\sin ^{2}x$$ or equivalent

$$\frac{5}{2}+2\sin 2x-\frac{3}{2}\cos 2x$$.

(ii)Integrate to obtain form $$px+q\cos 2x+r\sin 2x$$

Obtain $$\frac{5}{2}x-\cos 2x-\frac{3}{4} \sin 2x.$$

Substitute limits in integral of form  $$px+q\cos 2x+r\sin 2x$$ and attempt simplification

Obtain $$\frac{5}{8}\pi +\frac{1}{4}$$ or exact equivalent

Question

(i) Given that $$35+\sec ^{2}\Theta =12\tan \Theta$$, find the value of $$\tan \Theta$$.
(ii) Hence, showing the use of an appropriate formula in each case, find the exact value of
(a)$$\tan (\Theta -45^{\circ})$$

(b)$$\tan 2\Theta$$.

(i) Use $$\sec ^{2}\Theta=1+\tan ^{2}\Theta$$
Attempt solution of quadratic equation in $$\tan \Theta$$.
Obtain $$\tan ^{2}\Theta -12\tan \Theta +36=0$$ or equivalent and hence  $$\tan \Theta =6$$
(ii) (a) Attempt use of $$\tan (A-B)$$ formula
Obtain $$\frac{5}{7}$$  following their value of $$\tan \Theta$$
(b) Attempt use of $$\tan 2\Theta$$ formula
Obtain $$-\frac{12}{35}$$