Solving right triangles means finding missing side lengths and angle measures when one angle is 90 degrees. This skill matters because right triangles appear in ramps, shadows, navigation, construction, and physics components. The main tools are the Pythagorean theorem and the trigonometric ratios sine, cosine, and tangent.
With the right setup, a few known measurements can determine the entire triangle.
The key step is choosing a ratio based on the angle you are using and the sides you know or need. Relative to a chosen acute angle, the legs are called opposite and adjacent, while the longest side is the hypotenuse. Sine compares opposite to hypotenuse, cosine compares adjacent to hypotenuse, and tangent compares opposite to adjacent.
After solving for one missing part, you can often use 90 degrees minus the known acute angle to find the other acute angle.
Key Facts
- Pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse.
- sin(theta) = opposite / hypotenuse.
- cos(theta) = adjacent / hypotenuse.
- tan(theta) = opposite / adjacent.
- Acute angles in a right triangle add to 90 degrees: A + B = 90 degrees.
- Use inverse trig to find angles: theta = sin^-1(opposite / hypotenuse), cos^-1(adjacent / hypotenuse), or tan^-1(opposite / adjacent).
Vocabulary
- Right triangle
- A triangle with one angle measuring exactly 90 degrees.
- Hypotenuse
- The longest side of a right triangle, located across from the right angle.
- Opposite side
- The side across from the acute angle being used in a trigonometric ratio.
- Adjacent side
- The leg next to the acute angle being used, not including the hypotenuse.
- Inverse trigonometric function
- A function such as sin^-1, cos^-1, or tan^-1 used to find an angle from a side ratio.
Common Mistakes to Avoid
- Labeling opposite and adjacent before choosing an angle is wrong because those names depend on the reference angle.
- Using the hypotenuse as a leg in a^2 + b^2 = c^2 is wrong because c must always be the side across from the right angle.
- Choosing sine, cosine, or tangent by guessing is wrong because the correct ratio depends on which sides are known and which side or angle is missing.
- Leaving the calculator in radian mode for degree problems is wrong because it gives angle values in the wrong unit.
Practice Questions
- 1 A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
- 2 In a right triangle, an acute angle is 35 degrees and the hypotenuse is 12 m. Find the side opposite the 35 degree angle to the nearest tenth.
- 3 A student knows one leg and the hypotenuse of a right triangle and wants to find an acute angle. Explain whether sine, cosine, or tangent could be used, and what side labels must be checked first.