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 $$\frac{\sin \Theta }{\sin \Theta +\cos \Theta }+\frac{\cos \Theta }{\sin \Theta -\cos \Theta }=\frac{1}{\sin ^{2}\Theta -\cos ^{2}\Theta }$$.

(ii)Hence solve the equation $$\frac{\sin \Theta }{\sin \Theta +\cos \Theta }+\frac{\cos \Theta }{\sin \Theta -\cos \Theta }=3$$ for $$0^{\circ}\leq \Theta \leq 360^{\circ}$$

(i)$$\frac{\sin \Theta\left ( \sin \Theta -\cos \Theta \right )+\cos \Theta \left ( \sin \Theta +\cos \Theta \right ) }{\left ( \sin \Theta +\cos \Theta \right )\left ( \sin \Theta -\cos \Theta \right )}$$

$$\frac{\sin^{2}\Theta -\sin \Theta \cos \Theta +\sin \Theta \cos \Theta +\cos ^{2}\Theta }{\sin ^{2}\Theta -\cos ^{2}\Theta }$$

$$\frac{1}{\sin ^{2}\Theta -\cos ^{2}\Theta }$$

(ii)$$\sin ^{2}\Theta -\left ( 1-\sin ^{2} \Theta \right )=\frac{1}{3}$$ or $$1-\cos ^{2}\Theta -\cos ^{2}\Theta =\frac{1}{3}$$

or $$3\left ( \sin ^{2} \Theta -\cos ^{2}\Theta \right )=\cos ^{2}\Theta +\sin ^{2}\Theta$$

$$\sin \Theta =\pm \sqrt{\frac{2}{3}}$$ or $$\cos \Theta =\pm \sqrt{\frac{1}{3}}$$

$$\tan \Theta =\pm \sqrt{2}$$

$$\Theta =54.7^{\circ},125.3^{\circ},234.7^{\circ},305.3^{\circ}$$

Question

Solve the equation $$\sin 2x=2\cos 2x$$,for $$0^{\circ}\leq x\leq 180^{\circ}$$

$$\tan 2x=2$$

2x=63.4 or 243.4

x=31.7 or 121.7(allow 122)

Question

(i) Show that $$\cos ^{4}x=1-2\sin ^{2}x+\sin ^{4}x.$$

(ii)Hence,or otherwise,solve the equation $$8\sin ^{4}x+\cos ^{4}x=2\cos ^{2}x$$  for $$0^{\circ}\leq x\leq 360^{\circ}$$

(i)$$\cos ^{4}x=\left ( 1-\sin ^{2}x \right )^{2}=1-2\sin ^{2}x+\sin ^{4}x$$

(ii)$$8\sin ^{4}x+1-2\sin ^{2}x+\sin ^{4}x=2\left ( 1-\sin ^{2}x \right )$$

$$9\sin ^{4}x=1$$

$$x=35.3^{\circ}$$ (or any correct solution)

Any correct second solution from $$144.7^{\circ},215.3^{\circ},324.7^{\circ}$$

The remaining 2 solutions

Question

(i) Solve the equation 2 cos2θ = 3 sin θ, for 0◦ ≤ θ ≤ 360◦

(ii) The smallest positive solution of the equation $$2\cos ^{2}(n\Theta )=3\sin \left ( n\Theta \right )$$ , where n is a positive integer, is 10◦. State the value of n and hence find the largest solution of this equation in the interval 0◦ ≤ θ ≤ 360◦

(i)$$2(1-\sin ^{2}\Theta )=3\sin \Theta$$

$$(2\sin \Theta -1)(\sin \Theta +2)=0$$

$$\Theta =30^{\circ}$$ or $$150^{\circ}$$

(ii)$$n=\frac{their 30}{10}=3$$

(their 3)$$\Theta$$=720+their 150 =870

$$\Theta$$=$$290^{\circ}$$

.

Question

Solve the equation $$\frac{13\sin ^{2}\Theta }{2+\cos \Theta }=2$$  for $$0^{\circ}\leq \Theta \leq 180^{\circ}$$

$$13\sin ^{2}\Theta +2\cos \Theta +\cos ^{2}\Theta =4+2\cos \Theta$$

$$13\sin ^{2}\Theta +1-\sin ^{2}\Theta =4\rightarrow \sin ^{2}\Theta =\frac{1}{4}$$

or $$13-13\cos ^{2}\Theta +\cos ^{2}\Theta =4\rightarrow \cos ^{2}\Theta =\frac{3}{4}$$

$$30^{\circ}$$ , $$150^{\circ}$$

Question

(i) Solve the equation $$4\sin ^{2}x+8\cos x-7=0$$ for $$0^{\circ}\leq x\leq 360^{\circ}$$.

(ii) Hence find the solution of the equation $$4\sin ^{2}\left ( \frac{1}{2}\Theta \right )+8\cos \left ( \frac{1}{2} \Theta \right )-7=0$$  for  $$0^{\circ}\leq \Theta \leq 360^{\circ}$$

(i)$$4(1-\cos ^{2}x)+8\cos x-7=0$$

$$4c^{2}-8c+3=0\rightarrow \left ( 2\cos x-1 \right )(2\cos x-3 )=0$$

$$x=60^{\circ}$$ or $$300^{\circ}$$

(ii)$$\frac{1}{2}\Theta =60^{\circ}$$or $$300^{\circ}$$

$$\Theta =120^{\circ}$$only

#### Question.

(i) Prove the identity (ii) Hence solve the equation (i)$$\frac{1+\cos \Theta }{\sin \Theta }+\frac{\sin \Theta }{1+\cos \Theta }=\frac{2}{\sin \Theta }$$

$$\frac{\left ( 1+c \right )^{2}+s^{2}}{s\left ( 1+c \right )}=\frac{1+2c+c^{2+s^{2}}}{s(1+c)}$$

$$=\frac{2+2c}{s(1+c)}=\frac{2(1+c)}{s(1+c)}\rightarrow \frac{2}{s}$$

(ii)$$\frac{2}{s}=\frac{3}{c}\rightarrow t=\frac{2}{3}$$

$$\rightarrow \Theta =33.7^{\circ}or2134.7^{\circ}$$

#### Question. The diagram shows the graphs of y = tan x and y = cos x for $$0 ≤ x ≤ \Pi$$. The graphs intersect at points
A and B.
(i) Find by calculation the x-coordinate of A.

(ii) Find by calculation the coordinates of B.

5(i) $$\tan x =\cos x\rightarrow \sin x=\cos ^{2}x$$

$$\sin x=1-\sin ^{2}x$$  $$\sin x=0.6180$$ .Allow $$\frac{\left ( -1+\sqrt{5} \right )}{2}$$

x-cord of A=$$\sin ^{-1}0.618=0.666$$

(ii) EITHER

x-coordinate of B  is $$\pi -$$their 0.666

y-coordinate of B  is $$\tan \left ( their2.475 \right )$$ or $$\cos \left ( their 2.475 \right )$$

x=2.48,y=-0.786 or -0.787

OR

y-coordinate of B is-(cos or tan (their 0.66))

x-coordinate of B is $$\cos ^{-1 }$$( their y ) or $$\pi +\tan ^{-1}$$(their y )

x=2.48,y=-0.786 or -0.787

#### Question.

(a) Solve the equation $$sin^{-1}(3x) = -1$$, giving the solution in an exact form.
(b) Solve, by factorising, the equation $$2 cos\Theta sin\Theta – 2 cos\Theta – sin\Theta +1 = 0$$ for $$0 \leq \Theta \leq \Pi$$.

Ans:(a)$$(3x)=-\frac{\sqrt{3}}{2}\rightarrow x=\frac{-\sqrt{3}}{6}$$

(b)$$(2cos\Theta -1)(sin\Theta -1)=0$$
$$cos=\frac{1}{2} or sin\Theta =1$$
$$\Theta =\frac{\Pi }{3}$$ or $$\frac{\Pi }{2}$$

Question

(a) Solve the equation $$3sin^{2}2\Theta+8cos2\Theta =0$$ for 0Å ≤ 1 ≤ 180Å.

(b) The diagram shows part of the graph of y = a + tan bx, where x is measured in radians and a and
b are constants. The curve intersects the x-axis at $$(-\frac{\Pi }{ 6},0)$$and the y-axis at $$(0,\sqrt{3})$$ Find the
values of a and b.

(a)$$3(1-cos^{2}2\Theta)+8cos2\Theta =0\rightarrow 3cos^{2}2\Theta -8cos2\Theta -3(=0)$$

cos2θ$$=-\frac{1}{3}$$

2θ $$= 109.(47)o or 250.(53)o$$

θ = 54.7o or 125.3o

(b)
√3 tan0 3 = + a , a →=√3

0 tan( −bπ/ 6)  +√ 3 taken as far as  $$tan^{-1}$$, angle units consistent

b=2

### Question

(a) Solve the equation $$3tan^2 x-5 tan x – 2 =0$$ for $$0^o≤x≤180^0$$.
(b) Find the set of values of k for which the equation $$3tan^2 x – 5 tan x + k = 0$$ has no solutions.
(c) For the equation $$3tan^2 x – 5 tan x + k = 0$$, state the value of k for which there are three solutions in the interval $$0^o≤x≤180^o$$, and find these solutions.

Ans:

(a) $$(tan x-2)(3tanx+1)(=0)$$. or formula or completing square.
$$tan x =2 or -\frac{1}{3}$$
\)x=63.4^o\)(only value in range) or $$161.6^0$$ (only value in range)
(b) Apply $$b^2-4ac<0$$
$$k>\frac{25}{12}$$
(c) k = 0
tan x = 0 or $$\frac{5}{3}$$
$$x=0^0$$ or $$180^0or 59.0^0$$

### Question. The diagram shows part of the graph of y = a cos (bx) + c.

(a) Find the values of the positive integers a, b and c.

(b) For these values of a, b and c, use the given diagram to determine the number of solutions in the interval 0 ≤ x ≤ 2π for each of the following equations.

(i)   $$a cos (bx) + c =\frac{6}{\pi }x$$

(ii)   $$a cos (bx) + c = 6- \frac{6}{\pi }x$$

(a)  $$a = 5, b = 2 , c = 3$$

b.(i) 3

b.(i) 2

### Question

Solve the equation $$\frac{tan\Theta + 2sin\Theta }{tan\Theta – 2sin\Theta}=3for 0^o<\Theta <180^o$$.

Ans:

tan Θ+2sinΘ=3tanΘ-6sinΘ leading to 2tanΘ – 8sinΘ [=Θ]

$$cosΘ =\frac{1}{4}$$

$$Θ=75.5^o$$ only

### Question.

Solve, by factorising, the equation
6 cos θ tan θ − 3 cos θ + 4 tan θ − 2 = 0,
for 00 ≤ 1 ≤ 1800.

[leading to tan θ = $$\frac{1}{2}$$ , cos θ = $$-\frac{2}{3}$$ ]