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Difficulty: MediumSine and Cosine Rules

In ΔABC\Delta ABC, side a=10 cma = 10\text{ cm}, side b=16 cmb = 16\text{ cm}, and sin⁡A=38\sin A = \frac{3}{8}. If ∠B\angle B is an obtuse angle, what is the exact value of cos⁡B\cos B?

  1. −45-\frac{4}{5}Answer
  2. B
    45\frac{4}{5}
  3. C
    35\frac{3}{5}
  4. D
    −35-\frac{3}{5}

Answer

−45-\frac{4}{5}
Applying the Sine Rule gives sin⁡B=bsin⁡Aa=16×3810=35\sin B = \frac{b \sin A}{a} = \frac{16 \times \frac{3}{8}}{10} = \frac{3}{5}. Since ∠B\angle B is an obtuse angle, it lies in the second quadrant where cosine is negative. Using cos⁡B=−1−sin⁡2B\cos B = -\sqrt{1 - \sin^2 B}, we get cos⁡B=−1−(35)2=−45\cos B = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\frac{4}{5}.

Step-by-Step Solution

1
Apply the Sine Rule to calculate sin⁡B\sin B.
sin⁡B=bsin⁡Aa=16×3810=610=35\sin B = \frac{b \sin A}{a} = \frac{16 \times \frac{3}{8}}{10} = \frac{6}{10} = \frac{3}{5}
The Sine Rule states that asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}.
2
Determine the sign of cos⁡B\cos B based on the given angle type.
Since ∠B\angle B is an obtuse angle (90∘<B<180∘90^\circ < B < 180^\circ), cos⁡B<0\cos B < 0.
Cosine is negative in the second quadrant.
3
Calculate cos⁡B\cos B using the Pythagorean trigonometric identity.
cos⁡B=−1−sin⁡2B=−1−(35)2=−1625=−45\cos B = -\sqrt{1 - \sin^2 B} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\sqrt{\frac{16}{25}} = -\frac{4}{5}
Substitute sin⁡B=35\sin B = \frac{3}{5} into cos⁡B=−1−sin⁡2B\cos B = -\sqrt{1 - \sin^2 B}.

Key Concept

Sine Rule and Trigonometric Ratios of Obtuse Angles
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Topics

Sine and Cosine Rules

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