Use an Included Angle in a Nonright Triangle

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Two triangle sides are 2 and 3 cm with included angle 60°. Use cos(60°)=1/2 to find the exact opposite side. Candidate answers: (A) 5 cm (B) √1 cm (C) √26 cm (D) √7 cm.
• Independent check inputs: Two triangle sides are 3 and 4 cm with included angle 90°. Use cos(90°)=0 to find the exact opposite side. Candidate answers: (A) 7 cm (B) √1 cm (C) √25 cm (D) √50 cm.

Learn the idea

An identity holds for all inputs in its domain, while an equation asks for particular angle solutions. Algebraic identities, angle formulas and geometric laws must retain signs, intervals and their defining conditions. The law of cosines generalizes the right-triangle relationship by subtracting twice the side product times the included-angle cosine. A sixty-degree included angle has exact cosine one half.

Here is the method on the supplied teaching case. Law of cosines gives c²=2²+3²−2(2)(3)(1/2)=7. Therefore c=√7 cm. The included angle is opposite the requested side.

Match the included angle to the opposite requested side. The positive square root gives a length; the ordinary Pythagorean sum applies only when the included angle is ninety degrees and its cosine is zero.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Two triangle sides are 9 and 10 cm with included angle 90°. Use cos(90°)=0 to find the exact opposite side. Candidate answers: (A) 19 cm (B) √1 cm (C) √181 cm (D) √362 cm. Show the mathematical reason, not only the selected answer. Expected result: √181 cm. Law of cosines gives c²=9²+10²−2(9)(10)(0)=181. Therefore c=√181 cm. The included angle is opposite the requested side.

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