Proving angle or length value

A question is this type if and only if it requires showing that a specific angle or length equals a given exact value, typically using sine or cosine rule followed by algebraic manipulation.

4 questions · Challenging +1.3

1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05g Exact trigonometric values: for standard angles
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CAIE P1 2006 June Q6
6 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-2_389_995_1432_575} In the diagram, \(A B C\) is a triangle in which \(A B = 4 \mathrm {~cm} , B C = 6 \mathrm {~cm}\) and angle \(A B C = 150 ^ { \circ }\). The line \(C X\) is perpendicular to the line \(A B X\).
  1. Find the exact length of \(B X\) and show that angle \(C A B = \tan ^ { - 1 } \left( \frac { 3 } { 4 + 3 \sqrt { } 3 } \right)\).
  2. Show that the exact length of \(A C\) is \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).
CAIE P1 2016 June Q5
5 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-2_663_446_1562_847} In the diagram, triangle \(A B C\) is right-angled at \(C\) and \(M\) is the mid-point of \(B C\). It is given that angle \(A B C = \frac { 1 } { 3 } \pi\) radians and angle \(B A M = \theta\) radians. Denoting the lengths of \(B M\) and \(M C\) by \(x\),
  1. find \(A M\) in terms of \(x\),
  2. show that \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).
OCR H240/03 Q8
6 marks Challenging +1.2
8 In this question you must show detailed reasoning. The diagram shows triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-06_737_1383_456_342} The angles \(C A B\) and \(A B C\) are each \(45 ^ { \circ }\), and angle \(A C B = 90 ^ { \circ }\).
The points \(D\) and \(E\) lie on \(A C\) and \(A B\) respectively. \(A E = D E = 1 , D B = 2\). Angle \(B E D = 90 ^ { \circ }\), angle \(E B D = 30 ^ { \circ }\) and angle \(D B C = 15 ^ { \circ }\).
  1. Show that \(B C = \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 }\).
  2. By considering triangle \(B C D\), show that \(\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }\).
Edexcel AEA 2004 June Q7
19 marks Hard +2.3
Triangle \(ABC\), with \(BC = a\), \(AC = b\) and \(AB = c\) is inscribed in a circle. Given that \(AB\) is a diameter of the circle and that \(a^2\), \(b^2\) and \(c^2\) are three consecutive terms of an arithmetic progression (arithmetic series),
  1. express \(b\) and \(c\) in terms of \(a\), [4]
  2. verify that \(\cot A\), \(\cot B\) and \(\cot C\) are consecutive terms of an arithmetic progression. [3]
In an acute-angled triangle \(PQR\) the sides \(QR\), \(PR\) and \(PQ\) have lengths \(p\), \(q\) and \(r\) respectively.
  1. Prove that $$\frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R}.$$ [3]
Given now that triangle \(PQR\) is such that \(p^2\), \(q^2\) and \(r^2\) are three consecutive terms of an arithmetic progression,
  1. use the cosine rule to prove that $$\frac{2\cos Q}{q} = \frac{\cos P}{p} + \frac{\cos R}{r}.$$ [6]
  2. Using the results given in parts \((c)\) and \((d)\), prove that \(\cot P\), \(\cot Q\) and \(\cot R\) are consecutive terms in an arithmetic progression. [3]