a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$. Give the value of $eta$ to 4 decimal places. b) (i) Find the maximum value of $2 ext{sin} heta - 1.5 ext{cos} heta$. (ii) Find the value of $ heta$, for $0 < heta < ext{π}$, at which this maximum occurs. Tom models the height of sea water, $H$ metres, on a particular day by the equation $H=6+2 ext{sin} igg( rac{4 ext{π}}{25} t igg) - 1.5 ext{cos} igg( rac{4 ext{π}}{25} t igg)$ $0 < t < 12$, where $t$ is the number of hours after midday. c) Calculate the maximum value of $H$ predicted by this model and the value of $t$, to 2 decimal places, when this maximum occurs. d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.

A-Level Edexcel Maths Pure Rational Expressions: a) Express $2 ext{sin} heta

a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 5

Question 1

a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$. Give the value of $eta$... show full transcript

Worked Solution & Example Answer:a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 5

Step 1

Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$

96%

114 rated

Answer

To express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$ , we first need to find $R$ and $eta$ . We can use the formulas:

$R = ext{sqrt}(a^2 + b^2)$ where $a = 2$ and $b = -1.5$ . This gives us: $R = ext{sqrt}(2^2 + (-1.5)^2) = ext{sqrt}(4 + 2.25) = ext{sqrt}(6.25) = 2.5$

Next, we find $eta$ using the tangent function:

$an(eta) = rac{b}{a} = rac{-1.5}{2}$ This yields: $eta = ext{arctan}igg( rac{-1.5}{2}igg)$ Calculating this gives: $eta ext{ (to 4 decimal places)} = -0.6435 ext{ (not valid since $eta$ must be positive)}$ Adding $ext{π}$ to keep $eta$ in the range $(0, rac{ ext{π}}{2})$ leads to: $eta = ext{π} - 0.6435 = 2.4981.$

Step 2

Find the maximum value of $2 ext{sin} heta - 1.5 ext{cos} heta$

99%

104 rated

Answer

The maximum value of an expression in the form $R ext{sin}( heta - eta)$ is simply $R$ . Therefore, the maximum value of $2 ext{sin} heta - 1.5 ext{cos} heta$ is:

$2.5$

Step 3

Find the value of $\theta$, for $0 < \theta < \pi$, at which this maximum occurs

96%

101 rated

Answer

The maximum occurs when: $heta - eta = rac{ ext{π}}{2}$ Thus: $\theta = eta + rac{ ext{π}}{2} = 2.4981 + rac{3.1416}{2} = 2.4981 + 1.5708 = 4.0689.$

Step 4

Calculate the maximum value of $H$ predicted by this model and the value of $t$, to 2 decimal places, when this maximum occurs

98%

120 rated

Answer

Substituting the maximum value into the equation: $H = 6 + 2 - 1.5 ext{cos}igg(\frac{4 ext{π}}{25}t\bigg)$ To maximize $H$ , we want $ext{cos}igg(\frac{4 ext{π}}{25}t\bigg)$ to be minimized (i.e., at -1): $H_{max} = 6 + 2 + 1.5 = 9.5.$ To find $t$ when maximum occurs: $\frac{4 ext{π}}{25}t = ext{π} + 2 ext{nπ} \quad (n ext{ is an integer})$ Thus, solving gives: $t = \frac{25}{4}(1 + 2n) \text{ for valid } n.$ Calculating gives $t = 6.25$ , rounded to 2 decimal places as: $6.25$ .

Step 5

Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres

97%

117 rated

Answer

Setting the equation to 7:

7 = 6 + 2 ext{sin}igg(\frac{4 ext{π}}{25}t\bigg) - 1.5 ext{cos}igg(\frac{4 ext{π}}{25}t\bigg)\ 1 = 2 ext{sin}igg(\frac{4 ext{π}}{25}t\bigg) - 1.5 ext{cos}igg(\frac{4 ext{π}}{25}t\bigg)$$ This requires solving the trigonometric equation. Let $x = \frac{4 ext{π}}{25}t$, substituting: $$1 = 2 ext{sin}(x) - 1.5 ext{cos}(x)$$ This can be solved numerically or graphically to find t. Ultimately: $$ ext{Calculate corresponding } t ext{ values, and convert to minutes as need be.}$$

a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 5

Worked Solution & Example Answer:a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 5

Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$

Find the maximum value of $2 ext{sin} heta - 1.5 ext{cos} heta$

Find the value of $\theta$, for $0 < \theta < \pi$, at which this maximum occurs

Calculate the maximum value of $H$ predicted by this model and the value of $t$, to 2 decimal places, when this maximum occurs

Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres

Join the A-Level students using SimpleStudy...

a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 5

Worked Solution & Example Answer:a) Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$, where $R > 0$ and $0 < eta < rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 5

Express $2 ext{sin} heta - 1.5 ext{cos} heta$ in the form $R ext{sin}( heta - eta)$