Trigonometry

WMA11Topic 1.33 questions

Sine and cosine rules, radians, arc length, sector area and trigonometric graphs.

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Original practice questions written in the style of the exam.

Question 15 marks
In triangle $ABC$ , $AB = 7\,\text{cm}$ , $AC = 5\,\text{cm}$ and angle $BAC = 120^\circ$ .

(a) Find the exact length of $BC$ .

(b) Find the size of angle $ABC$ , giving your answer to 1 decimal place.
Mark scheme
- M1 Uses the cosine rule: $BC^2 = 7^2 + 5^2 - 2(7)(5)\cos 120^\circ$ (1)
Question 24 marks
A chord $PQ$ of a circle with centre $O$ and radius $8\,\text{cm}$ subtends an angle of $\dfrac{2\pi}{3}$ radians at .

−

2(7)(5)cos120∘

BC = \sqrt{109}\,\text{cm}

(1)

M1 Uses the sine rule with their

BC

\dfrac{\sin B}{5} = \dfrac{\sin 120^\circ}{\sqrt{109}}

(1)

\sin B = 0.4147...

(1)

A1 Angle

ABC = 24.5^\circ

(1)

2π

O

Find the exact area of the minor segment cut off by the chord $PQ$ .

Mark scheme

M1 Area of sector $= \tfrac{1}{2}(8^2)\left(\tfrac{2\pi}{3}\right) = \tfrac{64\pi}{3}$ (1)
M1 Area of triangle $OPQ = \tfrac{1}{2}(8^2)\sin\tfrac{2\pi}{3} = 16\sqrt{3}$ (1)
M1 Segment = sector − triangle (1)
A1 $\dfrac{64\pi}{3} - 16\sqrt{3}\,\text{cm}^2$ (1)

sin

32π

163

cm2