Properties Of Triangles

Question

Applying the sine rule in a triangle where angle $A = 30^\circ$, angle $B = 45^\circ$, and the side $a$ opposite $A$ measures $10$ units, the length of side $b$ equals which value?

RankGuru · Accepted Answer

The governing relation is the law of sines, which states that in any triangle the ratio of each side to the sine of its opposite angle is constant, written $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. This proportionality lets an unknown side be recovered as soon as one matching side-angle pair and the second angle are known. Rearranging the relevant proportion for $b$ gives $b = \frac{a \sin B}{\sin A}$. Substituting the standard values $\sin A = \sin 30^\circ = \frac{1}{2}$ and $\sin B = \sin 45^\circ = \frac{\sqrt{2}}{2}$ along with $a = 10$, we obtain $b = \frac{10 \cdot \frac{\sqrt{2}}{2}}{\frac{1}{2}} = \frac{5\sqrt{2}}{\frac{1}{2}} = 10\sqrt{2}$. Option $5\sqrt{2}$ forgets to divide by $\sin 30^\circ$, halving the result. Option $20$ incorrectly replaces $\sin 45^\circ$ with $1$. Option $10\sqrt{3}$ uses $\sin 60^\circ$ in place of $\sin 45^\circ$. This matches the canonical sine-rule application in JEE Advanced triangle problems. As a final plausibility check, the larger angle lies opposite the larger side, and since $B = 45^\circ > A = 30^\circ$, the side $b$ must exceed $a$; indeed $10\sqrt{2} \approx 14.1 > 10$, confirming the answer is geometrically consistent.

Properties Of Triangles

Select the correct option:

🔒 Solution Hidden from View

More trigonometry Practice Questions

In a triangle with sides $a$ , $b$ , $c$ and semiperimeter $s$ , the area computed by Heron's formula f...

In a triangle $A BC$ the sides satisfy $a = 13$ , $b = 14$ , and $c = 15$ units; using the cosine rule,...

Within a triangle, the identity relating angles states that $tan A + tan B + tan C$ equals which ...

In a triangle ABC, the sine rule is

In a triangle ABC, the cosine rule for side a is

Given that $tan^{- 1} x + tan^{- 1} y = \frac{π}{4}$ holds for positive reals with $x y < 1$ , the e...

If $cos θ = \frac{3}{5}$ with $θ$ in the first quadrant, then applying the half-angle iden...

The number of integer values of $k$ for which the equation $cos x = \frac{k}{3}$ admits at least on...

The fundamental period of the composite function $g (x) = sin^{4} x + cos^{4} x$ , after simplification ...

Statement based: assertion that $sin^{- 1} (sin \frac{2 π}{3}) = \frac{2 π}{3}$ and reason that arc...

By repeatedly using the triple-angle identity, the exact value of $sin 3 θ$ when $\sin\theta = ...

How many distinct values of $x$ in the open interval $(0, π)$ satisfy the equation $\sin x + \sqrt...

About This Question

Solution

Properties Of Triangles

Select the correct option:

🔒 Solution Hidden from View

More trigonometry Practice Questions

In a triangle with sides a, b, c and semiperimeter s, the area computed by Heron's formula f...

In a triangle ABC the sides satisfy a=13, b=14, and c=15 units; using the cosine rule,...

Within a triangle, the identity relating angles states that tanA+tanB+tanC equals which ...

In a triangle ABC, the sine rule is

In a triangle ABC, the cosine rule for side a is

Given that tan−1x+tan−1y=4π​ holds for positive reals with xy<1, the e...

If cosθ=53​ with θ in the first quadrant, then applying the half-angle iden...

The number of integer values of k for which the equation cosx=3k​ admits at least on...

The fundamental period of the composite function g(x)=sin4x+cos4x, after simplification ...

Statement based: assertion that sin−1(sin32π​)=32π​ and reason that arc...

By repeatedly using the triple-angle identity, the exact value of sin3θ when $\sin\theta = ...

How many distinct values of x in the open interval (0,π) satisfy the equation $\sin x + \sqrt...

About This Question

Solution

In a triangle with sides $a$ , $b$ , $c$ and semiperimeter $s$ , the area computed by Heron's formula f...

In a triangle $A BC$ the sides satisfy $a = 13$ , $b = 14$ , and $c = 15$ units; using the cosine rule,...

Within a triangle, the identity relating angles states that $tan A + tan B + tan C$ equals which ...

Given that $tan^{- 1} x + tan^{- 1} y = \frac{π}{4}$ holds for positive reals with $x y < 1$ , the e...

If $cos θ = \frac{3}{5}$ with $θ$ in the first quadrant, then applying the half-angle iden...

The number of integer values of $k$ for which the equation $cos x = \frac{k}{3}$ admits at least on...

The fundamental period of the composite function $g (x) = sin^{4} x + cos^{4} x$ , after simplification ...

Statement based: assertion that $sin^{- 1} (sin \frac{2 π}{3}) = \frac{2 π}{3}$ and reason that arc...

By repeatedly using the triple-angle identity, the exact value of $sin 3 θ$ when $\sin\theta = ...

How many distinct values of $x$ in the open interval $(0, π)$ satisfy the equation $\sin x + \sqrt...