Triangle with trigonometric identities

A question is this type if and only if it requires finding exact trigonometric values (like cos from sin, or using given exact ratios) before or during triangle calculations, often involving surds or fractions in the working.

4 questions · Moderate -0.5

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AQA C2 2012 June Q2
6 marks Moderate -0.8
2 The triangle \(A B C\), shown in the diagram, is such that \(A B = 26 \mathrm {~cm}\) and \(B C = 31.5 \mathrm {~cm}\). The acute angle \(A B C\) is \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\).
  1. Calculate the area of triangle \(A B C\).
  2. Find the exact value of \(\cos \theta\).
  3. Calculate the length of \(A C\).
Edexcel C2 Q4
5 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 6\) cm, \(BC = 4\) cm and \(CA = 5\) cm.
  1. Show that \(\cos A = \frac{3}{4}\). [3]
  2. Hence, or otherwise, find the exact value of \(\sin A\). [2]
OCR C2 Q4
8 marks Standard +0.3
\includegraphics{figure_4} In the diagram, \(ABCD\) is a quadrilateral in which \(AD\) is parallel to \(BC\). It is given that \(AB = 9\), \(BC = 6\), \(CA = 5\) and \(CD = 15\).
  1. Show that \(\cos BCA = -\frac{1}{3}\), and hence find the value of \(\sin BCA\). [4]
  2. Find the angle \(ADC\) correct to the nearest \(0.1°\). [4]
SPS SPS FM 2025 October Q2
6 marks Moderate -0.8
In a triangle \(ABC\), \(AB = 9\) cm, \(BC = 7\) cm and \(AC = 4\) cm.
  1. Show that \(\cos CAB = \frac{2}{3}\). [2]
  2. Hence find the exact value of \(\sin CAB\). [2]
  3. Find the exact area of triangle \(ABC\). [2]