1. ## AS.AQA.Page.220.Revision Exercise.Q8.Q19.Trigonometry.

Q8.
a)i)Express in sin^2 x in terms of cosX.
ii) By writing cosX=Y show that the equation 7cosX + 2 - 4 sin^2 x = 0 is equivalent to 4y^2 + 7y -2 = 0.

b) Solve the equation 4y^2 + 7y -2 = 0.

c)Hence solve the equation 7 cosX + 2 - 4 sin^2X = 0, giving all solutions to the nearest 0.1(degree) in the interval 0(degree)< x < 360(degree).

Q19.A student models the evening lighting-up time by the equation L = 6.125 - 2.25cos(πt/6) where the time, L hours pm, is always in GMT ( Greenwich Mean Time) and t is in months, starting in mid-December. The model assumes that all months are equally long.

a)Calculate the value of L for mid-January and for mid-May

b)Find, by solving an appropriate equation, the two months in the year when the lighting up time will be 5pm (GMT).

c)Write down an equation for L if t were to be in months starting in mid-March.

2. Originally Posted by ansonbound
Q8.
a)i)Express in sin^2 x in terms of cosX.
ii) By writing cosX=Y show that the equation 7cosX + 2 - 4 sin^2 x = 0 is equivalent to 4y^2 + 7y -2 = 0.

b) Solve the equation 4y^2 + 7y -2 = 0.

c)Hence solve the equation 7 cosX + 2 - 4 sin^2X = 0, giving all solutions to the nearest 0.1(degree) in the interval 0(degree)< x < 360(degree).

you should know sin^2 x + cos^2 x = 1
then 1-sin^2 x=cos^2 x

same for part ii)

can you do it now?

3. Originally Posted by ansonbound
Q8.
a)i)Express in sin^2 x in terms of cosX.
ii) By writing cosX=Y show that the equation 7cosX + 2 - 4 sin^2 x = 0 is equivalent to 4y^2 + 7y -2 = 0.

b) Solve the equation 4y^2 + 7y -2 = 0.

c)Hence solve the equation 7 cosX + 2 - 4 sin^2X = 0, giving all solutions to the nearest 0.1(degree) in the interval 0(degree)< x < 360(degree).

Q19.A student models the evening lighting-up time by the equation L = 6.125 - 2.25cos(πt/6) where the time, L hours pm, is always in GMT ( Greenwich Mean Time) and t is in months, starting in mid-December. The model assumes that all months are equally long.

a)Calculate the value of L for mid-January and for mid-May

b)Find, by solving an appropriate equation, the two months in the year when the lighting up time will be 5pm (GMT).

c)Write down an equation for L if t were to be in months starting in mid-March.
Hi ansonbound,

[Q8.a.i] $\sin^2 x = 1 - \cos^2 x$

[Q8.a.ii] $\cos x = y$

$7 \cos x + 2 - 4 \sin^2 x = 0 \equiv 4y^2+7y-2=0$

$7y+2-4(1-\cos^2 x)=0$

$7y+2-4+4 \cos^2x=0$

$7y-2+4y^2=0$

$4y^2+7y-2=0$

[Q8.b] $4y^2+7y-2=0$

$(4y-1)(y+2)=0$

$y=\frac{1}{4} \ \ or \ \ y=-2$

[Q8.c] $\boxed{\cos x =\frac{1}{4}}$

$\cos x = -2$ exceeds the range of cosine

$x=\{75.5, 284.5\}$

4. Thanks for Q.8, can someone help me out for Q19.?

5. (πt/6)
π is a "pie"