Trigonometry
Lessons in this chapter
About Trigonometry
Trigonometry builds from the Pythagorean Theorem into the full set of tools for solving any triangle, right or otherwise.
The Pythagorean Theorem and Special Right Triangles cover finding a right triangle’s missing side, including two special cases (45-45-90 and 30-60-90 triangles) with fixed side ratios, before Similar Right Triangles shows that the altitude to a right triangle’s hypotenuse creates two smaller triangles similar to the original.
The Tangent Ratio and The Sine and Cosine Ratios introduce the three core trigonometric ratios, each relating an acute angle in a right triangle to a specific pair of sides, and Real Life Applications of Trigonometry applies them to real problems like finding a height or distance that can’t be measured directly.
Law of Sines and Law of Cosines closes the chapter by extending trigonometric ratios beyond right triangles, solving for missing sides and angles in any triangle at all.
The seven lessons in this chapter each open as a Geometry worksheet in the page, including Special Right Triangles, The Sine and Cosine Ratios and Law of Sines and Law of Cosines. Every question is scored as you answer it, so picking sine where the given sides called for tangent is corrected on that problem instead of being repeated across a page of geometry problems. All seven are free to try, with no login needed for your first 3.
Frequently asked questions
The Pythagorean Theorem and Special Right Triangles find a missing side in a right triangle, including the 45-45-90 and 30-60-90 cases with fixed side ratios, and Similar Right Triangles shows that the altitude to the hypotenuse creates two triangles similar to the original. The Tangent Ratio and The Sine and Cosine Ratios introduce the three ratios that link an acute angle to a pair of sides, Real Life Applications of Trigonometry uses them for heights and distances that cannot be measured directly, and Law of Sines and Law of Cosines extends the work to triangles that are not right angled. The chapter sits in the high school Geometry conceptual category (G).
For an acute angle in a right triangle, sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.
Identify which two sides (relative to the given angle) are known or needed: opposite and hypotenuse call for sine, adjacent and hypotenuse call for cosine, and opposite and adjacent call for tangent.
The two legs are equal, and the hypotenuse is always the leg length multiplied by √2: a fixed ratio that doesn’t require the Pythagorean Theorem to calculate every time.
The side opposite the 30° angle is the shortest; the side opposite the 60° angle equals the shortest side times √3; and the hypotenuse (opposite the 90° angle) equals the shortest side times 2.
The Law of Sines relates each side of any triangle to the sine of its opposite angle in a fixed ratio, and it's used when you know an angle-side pair plus one more piece of information (like AAS or ASA information).