IB Maths AA SL 3.8 Trigonometric equations

[Maximum number: 12]

The following diagram shows a square ABCD , and a sector OAB of a circle centre O , radius r. Part of the square is shaded and labelled R.

Figure for Question IB Maths AA SL 3.8 Trigonometric Equations Questions question 1 — IB Maths AA SL

\mathrm{A} \hat{\mathrm{OB}}=\theta, \text { where } 0.5 \leq \theta<\pi

(a)

When $\theta=\alpha$ , the area of the square ABCD is equal to the area of the sector OAB .

[ 4 ]

(i) $\frac{1}{2} \alpha r^{2}\left(\right.$ accept $\left.2 r^{2}(1-\cos \alpha)\right)$

(i)

Hence find $\alpha$ .

[ 4 ]

correct equation in one variable
eg $2(1-\cos\alpha)=\frac{1}{2}\alpha$ $\alpha=0.511024$ $\alpha=0.511$ (accept $\theta=0.511$ )

2(1-\cos\alpha)=\frac{1}{2}\alpha,\quad \alpha=0.511

Note: Award A1 for $\alpha=0.511$ and additional answers.

(b)

When $\theta=\beta$ , the area of R is more than twice the area of the sector. Find all possible values of $\beta$ .

[ 8 ]

Note: In this part, accept $\theta$ instead of $\beta$ , and the use of equations instead of inequalities in the working.
attempt to find R
eg subtraction of areas, square - segment
correct expression for segment area
eg $\frac{1}{2} \beta r^{2}-\frac{1}{2} r^{2} \sin \beta$
correct expression for R
eg $\quad 2 r^{2}(1-\cos \beta)-\left(\frac{1}{2} \beta r^{2}-\frac{1}{2} r^{2} \sin \beta\right)$
correct inequality
eg $\quad 2 r^{2}(1-\cos \beta)-\left(\frac{1}{2} \beta r^{2}-\frac{1}{2} r^{2} \sin \beta\right)>2\left(\frac{1}{2} \beta r^{2}\right)$
correct inequality in terms of angle only
eg $\quad 2(1-\cos \beta)-\left(\frac{1}{2} \beta-\frac{1}{2} \sin \beta\right)>\beta$
attempt to solve their inequality, must represent R> twice sector eg sketch, one correct value

Note: Do not award the second (M1) unless the first (M1) for attempting to find R has been awarded. both correct values 1.30573 and 2.67369
correct inequality $1.31<\beta<2.67$

IB Maths AA SL 3.8 Trigonometric Equations Questions question 1

Question (a)

Question (i)

Question (b)