Challenging +1.2 Part (a) is a standard exact values problem using Pythagorean identity in a specified quadrant—routine for Further Maths students. Part (b) is more challenging, requiring knowledge of the cotangent sum formula for triangles and algebraic manipulation, but follows a recognizable pattern once the formula cot A + cot B + cot C = (a² + b² + c²)/(4Δ) is recalled or derived. Overall slightly above average difficulty due to part (b)'s non-standard application.
5. A cotangent function \(\cot x\) is defined as \(\cot x = \frac { \cos x } { \sin x } , x \neq 180 ^ { \circ } k , k \in \mathbb { Z }\).
a) If \(- 270 ^ { \circ } \leq \alpha \leq - 180 ^ { \circ }\) and \(\cot \alpha = - \frac { 12 } { 5 }\), find the exact value of \(\sin \alpha\) and \(\cos \alpha\).
b) If the sum of the squares of the side lengths of a triangle equals 2021 and the sum of the cotangents of its angles is 43 , find the area of that triangle. [0pt]
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5. A cotangent function $\cot x$ is defined as $\cot x = \frac { \cos x } { \sin x } , x \neq 180 ^ { \circ } k , k \in \mathbb { Z }$.\\
a) If $- 270 ^ { \circ } \leq \alpha \leq - 180 ^ { \circ }$ and $\cot \alpha = - \frac { 12 } { 5 }$, find the exact value of $\sin \alpha$ and $\cos \alpha$.\\
b) If the sum of the squares of the side lengths of a triangle equals 2021 and the sum of the cotangents of its angles is 43 , find the area of that triangle.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2022 Q5 [20]}}