3.8 Unit circle and trigonometric identitiesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A point $P$ on the unit circle, centre $O(0,0)$, has coordinates $(x,\,0.6)$ with $x<0$. The angle $\theta$ is measured anticlockwise from the positive $x$-axis to $OP$, where $0^\circ<\theta<180^\circ$.

   (a)

   Find the value of $\cos\theta$.

   [1 mark]

   • A$0.8$
   • B$-0.8$
   • C$-0.6$
   • D$0.64$

   (b)

   Find the value of $\tan\theta$.

   [1 mark]

   • A$\frac{3}{4}$
   • B$-\frac{4}{3}$
   • C$-\frac{3}{4}$
   • D$\frac{4}{3}$

   (c)

   Find the size of $\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A point $Q$ moves anticlockwise round the unit circle, starting at $(1,0)$. After turning through an angle $\theta$ from its start, $Q$ has coordinates $(\cos\theta,\sin\theta)$.

   (a)

   Find the coordinates of $Q$ when $\theta=210^\circ$, correct to 3 significant figures.

   [1 mark]

   • A$(-0.866,\,-0.5)$
   • B$(-0.5,\,-0.866)$
   • C$(0.866,\,-0.5)$
   • D$(-0.866,\,0.5)$

   (b)

   The graph of $y=\sin\theta$ is built by plotting the $y$-coordinate of $Q$ against $\theta$. For $0^\circ\le\theta\le360^\circ$, at which value of $\theta$ does the graph reach its minimum value?

   [1 mark]

   • A$180^\circ$
   • B$90^\circ$
   • C$360^\circ$
   • D$270^\circ$

   (c)

   Given that $\cos\theta=-0.28$ and that $Q$ is below the $x$-axis, find the value of $\sin\theta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In triangle $ABC$, $AB=12$ cm, $BC=9$ cm and $B\hat{A}C=40^\circ$.

   (a)

   Use the sine rule to find the two possible sizes of $A\hat{C}B$.

   [3 marks]

   (b)

   Find the two possible lengths of $AC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth of water, $d$ metres, in a harbour $t$ hours after midnight is modelled by $d=6+2.5\sin(30t^\circ)$, for $0\le t\le12$. Set your GDC to degree mode.

   (a)

   (i) Write down the maximum depth of the water.
   (ii) Find the period of the model.
   (iii) Use your GDC to find the times in $0\le t\le12$ at which the depth is $7.5$ m.

   [6 marks]

   (b)

   (i) Write down the first time at which the depth is a maximum.
   (ii) Use your GDC to find the times in $0\le t\le12$ at which the depth is $4$ m.
   (iii) Hence find, for $0\le t\le12$, the length of time for which the depth is less than $4$ m.

   [6 marks]

   Total for question 4: 12 marks

End of questions