Question
(i) Show that the equation \((\sqrt{2})cosecx+cotx =\sqrt{3}\)can be expressed in the form Rsin(x-\alpha )=\sqrt{2} where R > 0 and \(0Å < \alpha< 90Å.\) [4]
(ii) Hence solve the equation\( (\sqrt{2})\) cosec x + cot x = ï3, for 0Å < x < 180Å. [4]
Answer/Explanation
(i) Rearrange in the form State R =\( \sqrt{3}sinx-cox=\sqrt{2}\)
Use trig formulae to obtain α
Use trig formulae to obtain α
(ii) Evaluate\( sin^{-1}\left ( \frac{\sqrt{2}}{R} \right )\) Carry out a correct method to find a value of x in the given interval
Obtain answer x = 75°
Obtain a second answer e.g. x = 165° and no others
[Treat answers in radians as a misread. Ignore answers outside the given interval.]
Question
(i) By first expanding cos(2x + x), show that \(cos 3x \equiv 4cos^{3}x-3cosx\)
(ii) Hence solve the equation cos 3x + 3 cos x + 1 = 0, for 0 ≤ x ≤ \Pi.
(iii) Find the exact value of \(\int _{\frac{1}{6}\Pi }^{\frac{1}{3}\Pi }cos^{3}x dx\)
Answer/Explanation
(i) Use cos(A +B) formula to express cos3x in terms of trig functions of 2x and x Use double angle formulae and Pythagoras to obtain an expression in terms of cos x
only
Obtain a correct expression in terms of cos x in any form
Obtain \(cos3x= 4cos^{3}x-3cosx\)
(ii) Use identity and solve cubic \(4cos^{3}x=-1\) for x
Obtain answer 2.25 and no other in the interval
(iii) Obtain indefinite integral \(\frac{1}{12}sin3x+\frac{3}{4}sinx\)
Substitute limits in an indefinite integral of the form a xb x sin 3 sin + , where ab ≠ 0
Obtain answer \(\frac{1}{24}(\sqrt[9]{3-11})\) , or exact equivalent
Alternative method for question (iii)
\(\int cosx(1-sin^{2}x)dx=sinx-\frac{1}{3}sin^{3}x(+C)\)
Substitute limits in an indefinite integral of the form\( asinx+bsin^{3}x \)where ab ≠ 0
Obtain answer \( \frac{1}{24}(\sqrt[9]{3}-11)\) , or exact equivalent
Question
By first expressing the equation cot 1 −\( cot\Theta -cot(\Theta +45 )\)= 3 as a quadratic equation in tan 1, solve the equation for 0Å < 1 < 180Å.
Answer/Explanation
Use correct trig formula and obtain an equation in \( \tan \Theta \)
Obtain a correct horizontal equation in any form
Reduce to
Question
(i) Given that \(sin(\theta+45^{\circ})+2cos(\theta+60^{\circ})\) find the exact value of \(tan\theta \) in a form involving surds. You need not simplify your answer.
(ii) Hence solve the equation \(sin(\theta+45^{\circ})+2cos(\theta+60^{\circ})=3cos\theta \) for \(0^{\circ}<\theta <360^{\circ}\)
Answer/Explanation
(i)
Use trig formulae and obtain an equation in sin θ and cosθ M1 Obtain a correct equation in any form A1 Substitute exact trig ratios and obtain an expression for tanθ Obtain answer \(tanθ =\frac{\sqrt[2]{2}-1}{1-\sqrt{6}}\) , or equivalent
(ii)
State answer, e.g. θ = 128.4° B1 State second answer, e.g.θ = 308.4°