A ladder is standing on horizontal ground and leaning against a vertical wall. The length of the ladder is \(4.5\) metres. The distance between the bottom of the ladder and the base of the wall is \(2.2\) metres.

Use the above information to sketch a labelled diagram showing the ground, the ladder and the wall.

Calculate the distance between the top of the ladder and the base of the wall.

Calculate the obtuse angle made by the ladder with the ground.

(A1)   (C1)

Notes: Award (A1) for drawing an approximately right angled triangle, with correct labelling of the distances \(4.5\,({\text{m}})\) and \(2.2\,({\text{m}})\).

\(\sqrt {\,{{4.5}^2} - {{2.2}^2}} \,\,\,({\text{accept eqivalent }}eg\,\,{d^2} + {2.2^2} = {4.5^2})\)         (M1)

\( = 3.93\,({\text{m}})\,\,\,\left( {\sqrt {\,15.41} \,({\text{m}}),\,\,3.92555...\,({\text{m}})} \right)\)          (A1)    (C2)

Note: Award (M1) for a correct substitution in the Pythagoras formula.

\(180^\circ  - {\cos ^{ - 1}}\left( {\frac{{2.2}}{{4.5}}} \right)\)        (M1)(M1)

OR

\(180^\circ  - {\tan ^{ - 1}}\left( {\frac{{3.92555...}}{{2.2}}} \right)\)        (M1)(M1)

OR

\(180^\circ  - {\sin ^{ - 1}}\left( {\frac{{3.92555...}}{{4.5}}} \right)\)        (M1)(M1)

Note: Award (M1) for a correct substitution in the correct trigonometric ratio.
Award (M1) for subtraction from \(180^\circ \) (this may be implied if the sum of their inverse of the trigonometric ratio and their final answer equals \(180\)).

\( = 119^\circ \,\,\,(119.267...^\circ )\)        (A1)(ft)    (C3)

Note: Follow through from their part (b) if cosine is not used. Accept \(119.239...\) or \(119.151...\) from use of \(3\) sf values.

Question 3: Right angle trigonometry.
Candidates sketched the ladder leaning against the wall and recognized that Pythagoras’ theorem was needed to find the distance between the top of the ladder and the base of the wall (but not always correctly). Although it was a right triangle a number of the candidates used the law of sines (instead of Pythagoras’ theorem) and law of cosines (instead of a trigonometry ratio). Many candidates failed to find the obtuse angle made by the ladder with the ground even though the word obtuse was in bold type in the question.

Question 3: Right angle trigonometry.
Candidates sketched the ladder leaning against the wall and recognized that Pythagoras’ theorem was needed to find the distance between the top of the ladder and the base of the wall (but not always correctly). Although it was a right triangle a number of the candidates used the law of sines (instead of Pythagoras’ theorem) and law of cosines (instead of a trigonometry ratio). Many candidates failed to find the obtuse angle made by the ladder with the ground even though the word obtuse was in bold type in the question.

Question 3: Right angle trigonometry.
Candidates sketched the ladder leaning against the wall and recognized that Pythagoras’ theorem was needed to find the distance between the top of the ladder and the base of the wall (but not always correctly). Although it was a right triangle a number of the candidates used the law of sines (instead of Pythagoras’ theorem) and law of cosines (instead of a trigonometry ratio). Many candidates failed to find the obtuse angle made by the ladder with the ground even though the word obtuse was in bold type in the question.