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Question

Difficulty: MediumSine and Cosine Rules

In triangle XYZXYZ, side length x=3 cmx = 3\text{ cm}, side length y=8 cmy = 8\text{ cm}, and the included angle ∠Z=60∘\angle Z = 60^\circ. What is the length of side zz in cm?

Answer: 7 cm

Answer

The length of side zz is 7 cm7\text{ cm}.
Using the Cosine Rule z2=x2+y2−2xycos⁡Zz^2 = x^2 + y^2 - 2xy \cos Z, substituting x=3x = 3, y=8y = 8, and ∠Z=60∘\angle Z = 60^\circ yields z2=32+82−2(3)(8)(0.5)=9+64−24=49z^2 = 3^2 + 8^2 - 2(3)(8)(0.5) = 9 + 64 - 24 = 49. Taking the positive square root gives z=7 cmz = 7\text{ cm}.

Step-by-Step Solution

1
Identify known triangle components and select the appropriate rule
Two sides and the included angle (SAS) are given: x=3x = 3, y=8y = 8, ∠Z=60∘\angle Z = 60^\circ, requiring the Cosine Rule.
When given two sides and the included angle (SAS), the Cosine Rule is used to find the third side.
2
Substitute values into the Cosine Rule formula z2=x2+y2−2xycos⁡Zz^2 = x^2 + y^2 - 2xy \cos Z
z2=32+82−2(3)(8)cos⁡60∘z^2 = 3^2 + 8^2 - 2(3)(8)\cos 60^\circ
Direct algebraic substitution of side lengths and angle measure into the Cosine Rule.
3
Evaluate the trigonometric term and simplify the arithmetic expression
z2=9+64−48(0.5)=73−24=49z^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49
The exact value of cos⁡60∘\cos 60^\circ is 0.50.5.
4
Solve for side length zz by taking the principal square root
z=49=7 cmz = \sqrt{49} = 7\text{ cm}
Side length must be positive.

Key Concept

Applying the Cosine Rule to find the third side of a non-right-angled triangle given two sides and an included angle (SAS).
Estimated Time:1m 30s
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Topics

Sine and Cosine Rules

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