Trigonometric functions and circular measure - the Australian ...

A guide for teachers – Years 11 and 12 • {15}Suppose we are told that a triangle PQR has PQ = 9, ∠PQR = 45 ◦ and PR = 7. Then theangle opposite PQ is not uniquely determined. There are two non-congruent trianglesthat satisfy the given data.P97745 º θ'θQ R’ RApplying the sine rule to the triangle, we haveand sosinθ9= sin45◦7sinθ = 9sin45◦7≈ 0.9091.Thus θ ≈ 65 ◦ , assuming that θ is acute. But the supplementary angle is θ ′ ≈ 115 ◦ . Thetriangle PQR ′ also satisfies the given data. This situation is sometimes referred to as theambiguous case.Since the angle sum of a triangle is 180 ◦ , in some circumstances only one of the twoangles calculated is geometrically valid.The cosine ruleWe know from the SAS congruence test that a triangle is completely determined if we aregiven two sides and the included angle. However, if we know two sides and the includedangle in a triangle, the sine rule does not help us determine the remaining side or theremaining angles.The second important formula for general triangles is the cosine rule.