The area of a triangle = [tex] \frac{1}{2} [/tex] × the length of one side × the length of another side × sin(the angle between these two sides). For our example, because we know one side = 10 and the angle is 30° we will use these in our formula for 'easyness'.
Area of triangle = [tex] \frac{1}{2}bc \sin A[/tex] where A is the angle in-between the two sides b and c. (b=10, A = 30°, c=???)
We need to find c on your diagram and then we can use the formula above. We can use the sine rule to find the length of side c.
[tex] \frac{10}{\sin 45} = \frac{c}{\sin C} [/tex]
We know the angles in a triangle add up to 180° which means angle C must equal 180° - 45° - 30° = 105° so[tex] \frac{10}{\sin 45} = \frac{c}{\sin 105} \Rightarrow c= \frac{10}{\sin45} \times \sin105 = 13.66[/tex]
side c = 13.66 so
[tex]Area = \frac{1}{2}bc\sin \Rightarrow A = \frac{1}{2} \times 10 \times 13.66 \times \sin30=34.15[/tex]
Area = 34 square units (to nearest square unit)
