# SAT right triangles and trig: ratios and worked problems Source: https://1600.now/blog/sat-right-triangles-trig --- SAT Math Use the Pythagorean theorem, special triangles, sine, cosine, tangent, and complementary angles in SAT right-triangle questions. Written by [Luke Finigan](https://1600.now/about) 3 min read Updated Oct 2, 2026 Label the hypotenuse first. For a chosen acute angle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. ### The hypotenuse is opposite the right angle The hypotenuse is the longest side of a right triangle. It faces the $90^\circ$ angle; it isn't whichever side happens to look diagonal in the drawing. The other two sides are legs. For legs 6 and 8, $c^2=6^2+8^2=100$, so the hypotenuse is 10. If the hypotenuse is 13 and one leg is 5, the missing leg satisfies $b^2=13^2-5^2=144$, giving $b=12$. Adding the squares works when finding the hypotenuse. Finding a leg requires subtracting the known leg's square from the hypotenuse's square. Take the positive root because a side length is positive. Check that the hypotenuse remains the largest side. ### Scale the special triangles from the side you have In a $45^\circ$–$45^\circ$–$90^\circ$ triangle, the legs are equal and the hypotenuse is a leg times $\sqrt2$. If the hypotenuse is $12\sqrt2$, each leg is 12. The area is $\frac12(12)(12)=72$. In a $30^\circ$–$60^\circ$–$90^\circ$ triangle, write the sides as $s$, $s\sqrt3$, and $2s$. They face $30^\circ$, $60^\circ$, and $90^\circ$ respectively. If the long leg is $9\sqrt3$, then $s=9$ and the hypotenuse is 18. If instead the long leg is 9, the short leg is $9/\sqrt3=3\sqrt3$, and the hypotenuse is $6\sqrt3$. The given side determines the scale. Don't assume a labeled length is the short leg. ### Choose the ratio from the requested sides For acute angle $\theta$, $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$. SOH-CAH-TOA is a memory aid for these three definitions. In a 5–12–13 triangle, suppose the side opposite $\theta$ is 5. Then $\sin\theta=5/13$, $\cos\theta=12/13$, and $\tan\theta=5/12$. If you switch to the other acute angle, the two legs trade roles. When a question only asks for a ratio, you don't need to find the angle in degrees. Label the sides and divide. Calculating an inverse trig function adds work that the question hasn't requested. 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) - [Practice right triangles and trigonometry questions](https://1600.now/bank/math/skill/Right%20triangles%20and%20trigonometry) - [Print the right triangles and trigonometry worksheet](https://1600.now/sat-right-triangles-and-trig-worksheet) ### Recover a missing side from a trig ratio A right triangle has hypotenuse 20 and $\sin\theta=3/5$. Since sine is opposite divided by hypotenuse, $\dfrac{o}{20}=\dfrac35$, so the opposite side is 12. The adjacent side is $\sqrt{20^2-12^2}=16$. A ramp rises 4 feet and makes a $30^\circ$ angle with level ground. Its length is the hypotenuse, so $\sin30^\circ=4/L$. Since $\sin30^\circ=1/2$, $L=8$ feet. The horizontal distance would require the adjacent side, a different answer. If you enter trig expressions into a calculator, check whether the question's angle is in degrees or radians. The ratio definitions don't change, but the calculator must interpret the angle correctly. ### Use complementary angles without calculating either one The two acute angles in a right triangle sum to $90^\circ$. The side opposite one is adjacent to the other, so $\sin\theta=\cos(90^\circ-\theta)$. If $\sin A=7/25$ and B is the other acute angle, then $\cos B=7/25$. To find $\cos A$, recover the missing side: with opposite 7 and hypotenuse 25, adjacent is $\sqrt{625-49}=24$, so $\cos A=24/25$. The identity $\sin^2\theta+\cos^2\theta=1$ says the same thing through the Pythagorean theorem. In an acute right-triangle setting, both ratios are positive, so the positive square root gives the missing ratio. ### Draw the triangle the words describe For a ladder, its length is the hypotenuse, the wall height is a leg, and the ground distance is the other leg. For a rectangle's diagonal, the rectangle's width and height are legs. A sketch with these labels is enough; it doesn't need to be artistic. Use [right-triangle practice](https://1600.now/sat-skill/right-triangles-and-trig) to separate side-finding from ratio-finding. Then mix it with [geometry questions](https://1600.now/bank/math/browse). On review, name the side that your answer represents. If the question wants the ramp length and your work finds the horizontal distance, the arithmetic can be perfect while the answer is wrong. Luke Finigan is a student developer and the creator of 1600.now, a free Digital SAT practice platform. [About Luke](https://1600.now/about). ### Keep reading - **[SAT geometry: angles, area, similarity, circles, and trig](https://1600.now/blog/sat-geometry-overview)** A SAT geometry guide with worked angle, area, scale, volume, and coordinate examples, plus a practice order for the tested skills. - **[SAT Math formulas: reference sheet and what to learn](https://1600.now/blog/sat-math-formulas)** Compare Bluebook reference formulas with algebra, percent, circle, and trig formulas to learn, then practice choosing the right one. [Browse SAT resources →](https://1600.now/sat-resources)