3.10.4 Applications of Trigonometry
3.10.4 Applications of Trigonometry
3.10.4 Applications of trigonometry
It is critical to be able to determine which Rule or Formula to apply to answer a question.
The sine rule applies to triangles where you know the lengths of two sides and the size of the angle opposite one of those sides.
The cosine rule applies to triangles where you know the lengths of two sides and the size of the angle between them.
The sine rule is useful when you know the size of an angle and the lengths of two sides, and you need to find the length of a third side.
The cosine rule is useful when you know the lengths of two sides and the size of the angle between them, and you need to find the length of a third side or the size of an angle.
Worked example:
The diagram shows the positions of three points \( A, B \) and \( C \) in a field.
Show that \( BC \) is \( 118.1 \) m, correct to \( 1 \) decimal place.
We have two sides and angle between them and we are asked to show the length of the third side.
Here cosine rule will be helpful which is \( a^2=b^2+c^2-2 \ bc \ Cos(A) \)
Let’s label the diagram
Substitute values in the formula
\( (BC)^2=115^2+80^2-2\times 115\times 80\times cos (72) \)
\( (BC)^2=13939.0873 \)
\( BC = \sqrt{13939.0873} \)
\( BC = 118.06239119… \)
Rounding to one decimal place
\( BC = 118.1 \)
\( BC = 118.1 \)m
Calculate angle \( ABC \)
Now we know all three sides and an angle, so to find angle ABC we use sine rule, which is \( \frac{a}{sin A} = \frac{b}{sin B} \)
Substitute the values from the diagram
\( \frac{118.1}{sin 72} = \frac{115}{sin B} \)
Rearrange the equation
\( sin B = \frac{115\times sin(72) }{118.1} \)
\( B = \frac{115\times sin(72)}{118.1} \)
\( B=67.83364349 \)
Rounding to one decimal place
\( B=67.8 \)
Angle \( ABC=67.8° \)
Test yourself
Question 1:
The diagram shows two triangles.
Calculate \( QR \).
[3]
Calculate \( RS \).
[4]
Calculate the total area of the two triangles.
[3]
Question 2:
Use the cosine rule to find angle \( BAC \).
[4]
Question 3:
The diagram shows a field, \( ABCD \), on a horizontal ground.
Use the cosine rule to find angle \( BAC \).
[4]
Use the sine rule to find angle \( CAD \).
[3]
Calculate the area of the field.
[3]
Question 4:
Calculate angle \( ACB \).
[4]
Calculate angle \( ACD \).
[4]
Calculate area of the quadrilateral \( ABCD \).
[3]
Question 5:
In the pentagon \( ABCDE \), angle \( ACB = \)angle \( AED=90° \).
Triangle \( ACD \) is equilateral with side length \( 12 \) cm.
\( DE=BC=6 \) cm.
Calculate angle \( BAE \).
[4]
Calculate the area of the pentagon.
[4]
Calculate \( AB \).
[2]
Calculate \( AE \).
[3]
Question 6:
The diagram shows a quadrilateral \( PQRS \) formed from two triangles, \( PQS \) and \( QRS \).
Calculate
\( QR \),
[3]
\( PS \),
[3]
The area of quadrilateral \( PQRS \).
[4]