Right Triangles and Trigonometry

Put the hypotenuse on the correct side of the equation

The hypotenuse is opposite the right angle and is the longest side. For legs 5 and 12, $c^2=5^2+12^2=169$, so $c=13$. If 13 is given as the hypotenuse and 5 as a leg, the other leg satisfies $b^2=13^2-5^2$.

Only right triangles support $a^2+b^2=c^2$. A triangle's appearance is not enough; use a right-angle marking or a stated perpendicular relationship.

Label sides relative to the requested angle

For angle $\theta$, $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$. In a 5–12–13 triangle, an angle opposite the side 5 has sine $\frac{5}{13}$.

Switching to the other acute angle swaps opposite and adjacent. The hypotenuse stays the same. This is why complementary acute angles satisfy $\sin \theta =\cos (90^{\circ }-\theta )$.

Scale special triangles from the known side

A 45–45–90 triangle has sides $s,s,s\sqrt{2}$. A 30–60–90 triangle has sides $s,s\sqrt{3},2s$, where $s$ is opposite the 30-degree angle. If its hypotenuse is 10, the short leg is 5 and the long leg is $5\sqrt{3}$.

Identify which side was supplied before multiplying. A long leg of 10 in a 30–60–90 triangle would give $s=\frac{10}{\sqrt{3}}$, not 5.

Check angle units and the requested quantity

If you evaluate a trig expression with a calculator, use the mode that matches degrees or radians in the stem. If the question gives a ratio and asks for another ratio, side relationships may avoid finding the angle at all.

Keep exact radicals or fractions when requested. A decimal estimate can help check the size, but it should not replace an exact expression in the final answer.

Check your understanding

Sample right triangles and trigonometry questions

Practice this skill

1. Mark the right angle and hypotenuse on every diagram.
2. Label opposite and adjacent afresh for each requested angle.
3. Practice identifying the known side in both special triangle families.

Print the right triangles and trigonometry worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.