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Zorluk: KolaySine and Cosine Rules

In ΔABC\Delta ABC, the length of side aa is 10 cm10\text{ cm}, angle ∠A=30∘\angle A = 30^\circ, and angle ∠B=45∘\angle B = 45^\circ. What is the length of side bb?

  1. 102 cm10\sqrt{2}\text{ cm}Cevap
  2. B
    52 cm5\sqrt{2}\text{ cm}
  3. C
    103 cm10\sqrt{3}\text{ cm}
  4. D
    20 cm20\text{ cm}

Cevap

The length of side bb is 102 cm10\sqrt{2}\text{ cm}.
The option specifying 102 cm10\sqrt{2}\text{ cm} is correct because applying the Sine Rule gives 10sin⁡30∘=bsin⁡45∘\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}, which simplifies directly to b=102 cmb = 10\sqrt{2}\text{ cm}.

Adım Adım Çözüm

1
State the Sine Rule formula relating sides aa, bb and their opposite angles AA, BB.
asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}
The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.
2
Substitute the known values (a=10a = 10, A=30∘A = 30^\circ, B=45∘B = 45^\circ) into the formula.
10sin⁡30∘=bsin⁡45∘\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}
Isolating the variable bb requires substituting given numeric angle and side measures.
3
Evaluate exact values of trigonometric functions and solve for bb.
b=10×2212=102 cmb = \frac{10 \times \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2}\text{ cm}
Simplifying the algebraic fraction yields the exact length of side bb.

Anahtar Kavram

Application of the Sine Rule to find an unknown side length in a non-right-angled triangle.
Tahmini Süre:45s
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Konular

Sine and Cosine Rules

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