In \Delta ABC, side a = 4\text{ cm}, side b = 4\sqrt{2}\text{ cm}, and \angle A = 30^\circ… — JAMB UTME

In $\Delta ABC$ , side $a = 4\text{ cm}$ , side $b = 4\sqrt{2}\text{ cm}$ , and $\angle A = 30^\circ$ . If $\angle B$ is an obtuse angle, what is the measure of $\angle B$ ?

A
$45^\circ$
$135^\circ$ Cevap
C
$105^\circ$
D
$150^\circ$

Cevap

135^\circ

Applying the Sine Rule gives

\frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin B}

, which simplifies to

\sin B = \frac{\sqrt{2}}{2}

. The inverse sine operation yields an acute angle of

45^\circ

and an obtuse angle of

180^\circ - 45^\circ = 135^\circ

. Since the stem specifies that angle B is obtuse, the correct value is

135^\circ

Adım Adım Çözüm

Apply the Sine Rule formula relating sides

a

b

and their opposite angles

A

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.

Substitute the known values into the equation:

a = 4

b = 4\sqrt{2}

, and

A = 30^\circ

\frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin B} \implies \frac{4}{0.5} = \frac{4\sqrt{2}}{\sin B} \implies 8 = \frac{4\sqrt{2}}{\sin B}

\sin 30^\circ = 0.5

Solve for

\sin B

\sin B = \frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2}

Rearranging the equation yields the value for

\sin B

Find the obtuse angle whose sine is

\frac{\sqrt{2}}{2}

\angle B = 180^\circ - 45^\circ = 135^\circ

Sine is positive in both the first and second quadrants. The acute reference angle is

\arcsin\left(\frac{\sqrt{2}}{2}\right) = 45^\circ

, so the supplementary obtuse angle is

180^\circ - 45^\circ = 135^\circ

Anahtar Kavram

Sine Rule and the Ambiguous Case

Soru

Cevap

Adım Adım Çözüm

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