Finding Missing Sides

A right-angled triangle has θ = $30$° and hypotenuse = $10$. Find the opposite side.

Step $1$ — label. Hypotenuse = $10$ (known). Opposite = ? (wanted). Adjacent is not needed.

Step $2$ — choose. Known H, want O → use sin (SOH).

Step $3$ — write the ratio. $\sin (30^{\circ })=\frac{\text{Opposite}}{10}$

Step $4$ — look up sin($30$°) in the table: $\sin (30^{\circ })=\frac{1}{2}$.

Step $5$ — rearrange (multiply both sides by $10$): $\text{Opposite}=10\times \frac{1}{2}=5$

Answer: $5$

A right-angled triangle has θ = $60$° and hypotenuse = $8$. Find the adjacent side.

Step $1$ — label. Hypotenuse = $8$. Adjacent = ?

Step $2$ — choose. Known H, want A → use cos (CAH).

Step $3$ — write: $\cos (60^{\circ })=\frac{\text{Adjacent}}{8}$

Step $4$ — table: $\cos (60^{\circ })=\frac{1}{2}$.

Step $5$ — rearrange: $\text{Adjacent}=8\times \frac{1}{2}=4$

Answer: $4$

A right-angled triangle has θ = $40$° and adjacent = $12$. Find the opposite side.

Step $1$ — label. Adjacent = $12$. Opposite = ?

Step $2$ — choose. Known A, want O → use tan (TOA).

Step $3$ — write: $\tan (40^{\circ })=\frac{\text{Opposite}}{12}$

Step $4$ — calculator (degree mode): $\tan (40^{\circ })\approx 0.8391$

Step $5$ — rearrange: $\text{Opposite}=12\times 0.8391\approx 10.07$

Answer: ≈ $10.07$

A right-angled triangle has θ = $30$° and opposite = $6$. Find the hypotenuse.

Step $1$ — label. Opposite = $6$. Hypotenuse = ?

Step $2$ — choose. Known O, want H → use sin.

Step $3$ — write: $\sin (30^{\circ })=\frac{6}{\text{Hypotenuse}}$

Step $4$ — table: $\sin (30^{\circ })=\frac{1}{2}$, so $\frac{1}{2}=\frac{6}{\text{Hypotenuse}}$

Step $5$ — rearrange (multiply both sides by the hypotenuse, then by $2$): $\text{Hypotenuse}=\frac{6}{\sin (30^{\circ })}=\frac{6}{\frac{1}{2}}=12$

Answer: $12$