Sine and cosine rules and triangle area

Geometry and measures • Mensuration and calculation

Flashcards

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What is the preferred method when both a side and its opposite angle are missing?

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Identify any solvable pair; prefer the cosine rule if the included angle is known, otherwise use the sine rule when an opposite angle is available.

Key concepts

What you'll likely be quizzed about

Sine rule: statement and use

The sine rule states a/sin A = b/sin B = c/sin C, where a, b, c denote side lengths opposite angles A, B, C respectively. The rule follows from equal ratios of side to sine of opposite angle in any triangle, because corresponding altitudes produce proportional sines. The sine rule suits cases with two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

Sine rule: solving for unknowns and the ambiguous case

Rearrangement of the sine rule isolates the required quantity: a = (sin A)·(b/sin B) or sin A = a·sin B / b. Calculation of angles from sine values requires attention to the range of sine: sin θ = k may give θ or 180°−θ (the ambiguous case). The ambiguous case occurs when two sides and a non-included angle are known (SSA) and can produce zero, one, or two valid triangles depending on the computed sine value and side lengths.

Cosine rule: statement and use

The cosine rule states a^2 = b^2 + c^2 − 2bc·cos A, with variations for each side: b^2 = a^2 + c^2 − 2ac·cos B, etc. The rule reduces to Pythagoras when the included angle equals 90°. The cosine rule suits cases with two sides and the included angle (SAS) to find the third side, or with three sides (SSS) to find an angle via cos A = (b^2 + c^2 − a^2)/(2bc).

Area formula for any triangle

The general area formula Area = 1/2·a·b·sin C uses two sides and the included angle between them. The formula follows because area equals 1/2·base·height and the height equals b·sin C when base = a. The formula applies to any triangle, including obtuse cases, because sin of the included angle gives the perpendicular component of one side onto the other.

Key notes

Important points to keep in mind

Label sides and angles clearly with lowercase for sides and uppercase for opposite angles (a opposite A).

Identify known quantities first; classify the case as ASA/AAS, SAS, SSS or SSA to choose the appropriate formula.

Use the sine rule for angle–side pairs and the cosine rule for cases involving the included angle or three sides.

Check for the ambiguous SSA case by comparing the given side to the computed height b·sin A.

Compute angles with inverse cosine when three sides are known to avoid ambiguity from sine.

When using Area = 1/2·a·b·sin C, ensure C is the included angle between sides a and b.

Reduce the cosine rule to Pythagoras when the included angle is 90° to check calculations.

Always verify that computed angles sum to 180° and that derived sides are positive and satisfy triangle inequalities.