# Right Triangles and Trigonometry on the SAT: Examples & Practice Source: https://1600.now/sat-skill/right-triangles-and-trig --- Math · Geometry and Trigonometry ## Right Triangles and Trigonometry Locate the right angle first. Use the Pythagorean theorem for side lengths, and label opposite, adjacent, and hypotenuse relative to the angle named in the question before using a trig ratio. Written by [Luke Finigan](https://1600.now/about) 2 min read Updated Oct 2, 2026 ### 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 $\frac5{13}$. 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 $s,s,s\sqrt2$. A 30–60–90 triangle has sides $s,s\sqrt3,2s$, where $s$ is opposite the 30-degree angle. If its hypotenuse is 10, the short leg is 5 and the long leg is $5\sqrt3$. Identify which side was supplied before multiplying. A long leg of 10 in a 30–60–90 triangle would give $s=\frac{10}{\sqrt3}$, 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 Try it yourself A right triangle has legs 6 and 8 and hypotenuse 10. What is the sine of the angle opposite the side of length 6? A $\frac34$ B $\frac35$ C $\frac45$ D $\frac43$ ### Sample right triangles and trigonometry questions Right Triangles and Trigonometry · Easy Which expression represents the length of line segment $AB$? A $14\sin 58^\circ$ B $\frac{14}{\sin 58^\circ}$ C $14\cos 58^\circ$ D $\frac{14}{\cos 58^\circ}$ - [Open the question and explanation](https://1600.now/bank/math/5fadad1b) Right Triangles and Trigonometry · Easy The measure of angle B is $\frac{2\pi}{3}$ radians. What is the value of $sin(B)$? A $-\frac{\sqrt{2}}{2}$ B $-\frac{\sqrt{3}}{2}$ C $\frac{1}{2}$ D $\frac{\sqrt{3}}{2}$ - [Open the question and explanation](https://1600.now/bank/math/416b066a) Right Triangles and Trigonometry · Easy The measure of angle $C$ is $\frac{5\pi}{6}$ radians. What is the value of $\cos(C)$? A $-\frac{\sqrt{3}}{2}$ B $-\frac{1}{2}$ C $\frac{\sqrt{3}}{2}$ D $\frac{1}{2}$ - [Open the question and explanation](https://1600.now/bank/math/fd3389f2) ### 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. [Practice right triangles and trigonometry](https://1600.now/bank/math/skill/Right%20triangles%20and%20trigonometry) [Print the right triangles and trigonometry worksheet and worked answers](https://1600.now/sat-right-triangles-and-trig-worksheet). Then use a [mixed practice module](https://1600.now/modules) to check whether you can choose this method among other question types. ### Source and question labels The bank label for this guide is "Right triangles and trigonometry" in Geometry and Trigonometry. The worked examples above are written for this guide. - [College Board: Types of Math tested](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types) ### Related skills - **[Lines, Angles, and Triangles](https://1600.now/sat-skill/lines-angles-triangles)** Name the geometric relationship before writing an equation. - **[Circles](https://1600.now/sat-skill/circles)** Read a circle's radius before using area or circumference. - **[Area and Volume](https://1600.now/sat-skill/area-and-volume)** Choose the formula for the requested quantity, then identify its perpendicular height and units.