Q.In a right-angled triangle, the side opposite to angle $\theta$ measures 3 units, the side adjacent to $\theta$ measures 4 units, and the hypotenuse measures 5 units. Find the values of all six trigonometric ratios of $\theta$.

Each ratio is a simple fraction of the two given sides against the third, taken straight from the six ratio definitions.

$\sin\theta = \dfrac{3}{5}$, $\cos\theta = \dfrac{4}{5}$, $\tan\theta = \dfrac{3}{4}$, and their reciprocals $\text{cosec}\,\theta = \dfrac{5}{3}$, $\sec\theta = \dfrac{5}{4}$, $\cot\theta = \dfrac{4}{3}$.

Step 1 — identify the sides. Opposite to $\theta$ = 3, adjacent to $\theta$ = 4, hypotenuse = 5. (Check: $3^2+4^2=9+16=25=5^2$, so this is a valid right triangle.)

Step 2 — apply the three basic ratio definitions.

Step 3 — take reciprocals for the remaining three.

Students often swap the opposite and adjacent sides when the angle $\theta$ is not explicitly marked in the figure, which silently swaps $\sin\theta$ with $\cos\theta$ and $\tan\theta$ with $\cot\theta$. Always identify which side is across from $\theta$ (opposite) versus which touches $\theta$ without being the hypotenuse (adjacent) before writing any ratio.