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 \(0^{\circ}\leq x\leq 360^{\circ}\).
Answer/Explanation
(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.
Answer/Explanation
(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 \(0^{\circ}\leq \Theta \leq 360^{\circ}\)
Answer/Explanation
(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 \(0^{\circ}\leq x\leq 180^{\circ}\)
Answer/Explanation
\(\tan 2x=2\)
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 \(0^{\circ}\leq x\leq 360^{\circ}\)
Answer/Explanation
(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◦
Answer/Explanation
(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 \(0^{\circ}\leq \Theta \leq 180^{\circ}\)
Answer/Explanation
\(`13\sin ^{2}\Theta +2\cos \Theta +\cos ^{2}\Theta =4+2\cos \Theta\)
\(`13\sin ^{2}\Theta +1-\sin ^{2}\Theta =4\rightarrow \sin ^{2}\Theta =\frac{1}{4}\)
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 \(0^{\circ}\leq \Theta \leq 360^{\circ}\)
Answer/Explanation
(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 
Answer/Explanation
(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.
Answer/Explanation
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 \(0 \leq \Theta \leq \Pi\).
Answer/Explanation
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.
Answer/Explanation
(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.
Answer/Explanation
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) \(a cos (bx) + c = 6- \frac{6}{\pi }x\)
Answer/Explanation
(a) \(a = 5, b = 2 , c = 3\)
b.(i) 3
b.(i) 2
Question
Solve the equation \(\frac{tan\Theta + 2sin\Theta }{tan\Theta – 2sin\Theta}=3for 0^o<\Theta <180^o\).
Answer/Explanation
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.
Answer/Explanation
3 cos θ (2 tan θ-1) + 2(2tanθ-1) [=0]
(2 tan θ – 1) (3 cos θ+2) [=0]
[leading to tan θ = \(\frac{1}{2}\) , cos θ = \(-\frac{2}{3}\) ]
26.60, 131.80