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Difficulty: EasySine and Cosine Rules

In ΔPQR\Delta PQR, side p=3 cmp = 3\text{ cm}, side q=8 cmq = 8\text{ cm}, and the included angle ∠R=60∘\angle R = 60^\circ. What is the length of side rr in cm?

Answer: 7 cm

Answer

The length of side rr is 7 cm7\text{ cm}.
Using the Cosine Rule r2=p2+q2−2pqcos⁡Rr^2 = p^2 + q^2 - 2pq \cos R with p=3p=3, q=8q=8, and ∠R=60∘\angle R=60^\circ yields r2=9+64−48(0.5)=49r^2 = 9 + 64 - 48(0.5) = 49, which gives r=7 cmr = 7\text{ cm}.

Step-by-Step Solution

1
State the Cosine Rule formula for side rr
r2=p2+q2−2pqcos⁡Rr^2 = p^2 + q^2 - 2pq \cos R
The Cosine Rule allows calculating the third side of a triangle when two sides and the included angle are given.
2
Substitute the given values into the formula
r2=32+82−2(3)(8)cos⁡60∘r^2 = 3^2 + 8^2 - 2(3)(8) \cos 60^\circ
We are given p=3p = 3, q=8q = 8, and ∠R=60∘\angle R = 60^\circ.
3
Evaluate the trigonometric expression and simplify
r2=9+64−48(0.5)=73−24=49r^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49
Since cos⁡60∘=0.5\cos 60^\circ = 0.5, multiplying 2×3×8×0.52 \times 3 \times 8 \times 0.5 yields 2424.
4
Solve for rr by taking the positive square root
r=7 cmr = 7\text{ cm}
Length must be a positive value, and 49=7\sqrt{49} = 7.

Key Concept

Cosine Rule for finding an unknown side given two sides and the included angle (SAS).
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Topics

Sine and Cosine Rules

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