IB Maths AI HL 3.8 Unit circle and trigonometric equations

[Maximum number: 6]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

Figure for Question IB Maths AI HL Ahl 3 8 Hl Unit Circle and Trigonometric Equations Questions question 1 — IB Maths AI HL

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

Show that the maximum value of $\omega=1.98$ , correct to three significant figures.

[ 3 ]

substitution of either value of $\delta$ into equation

Note: Both (M1) lines may be seen in either part (c)(i) or part (c)(ii).

\cos \omega=0.839 \tan (-0.440)

\omega=1.97684 \ldots

\approx 1.98

Note: For substitution of 1.98 award MOAO.

(b)

Find the minimum value of $\omega$ .

[ 1 ]

$\delta=0.440$

(c)

Hence show that a=1.6, correct to two significant figures.

[ 2 ]

$a=\frac{7.76075 \ldots-4.57258 \ldots}{2}$ $\approx 1.59408 \ldots$

Note: Award M1 for substituting their values into a correct expression.

Award A1 for a correct value of a from their expression which has at least 3 significant figures and rounds correctly to 1.6 .
$\approx 1.6$ (correct to 2 sf )

IB Maths AI HL Ahl 3 8 Hl Unit Circle and Trigonometric Equations Questions question 1

Question (a)

Question (b)

Question (c)