3.51: Trigonometric function form, equation and gradient

0606/23/M/J/24 — Question 5 · 14 marks

(a)(i)

The function

\mathrm{f}

is defined by

\mathrm{f}(x) = \dfrac{1 + 2\mathrm{sin}^{2} x}{\mathrm{cos}^{2} x}

for

-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}

.

Show that

\mathrm{f}(x)

can be written as

a\,\mathrm{tan}^{2} x + b

, where

a

and

b

are integers.

[2]

(a)(ii)

Hence solve the equation

\mathrm{f}(x) = 4

.

[3]

(a)(iii)

Hence also find the gradient of the curve

y = \mathrm{f}(x)

at each of the points where

y = 4

.

[4]

(b)

Solve the equation

50\,\mathrm{cos}^{2}\theta = 5\,\mathrm{sin}\,\theta + 47

for

0^{\circ} \leqslant \theta \leqslant 360^{\circ}

.

[5]