Right triangles

Learning Objectives Identify the hypotenuse, opposite, and adjacent sides of a right triangle relative to a given acute angle. Apply the Pythagorean theorem and its converse to solve for unknown side lengths. Use the properties of 45-45-90 and 30-60-90 special right triangles to find exact side lengths without a calculator. Define and calculate the sine, cosine, and tangent of an acute angle in a right triangle. Use trigonometric ratios (sin, cos, tan) to find unknown side lengths in right triangles. Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find unknown angle measures in right triangles. Model and solve real-world problems involving angles of elevation and depression. Ever wondered how surveyors measure the height of a massive skyscraper or a mountain w...

TermDefinitionExample Right TriangleA triangle that has one angle measuring exactly 90 degrees. The other two angles are always acute (less than 90 degrees).A triangle with angles 30°, 60°, and 90° is a right triangle. HypotenuseThe side of a right triangle that is directly opposite the right angle. It is always the longest side of the triangle.In a triangle with sides 3, 4, and 5, the side with length 5 is the hypotenuse because it is opposite the right angle. LegsThe two sides of a right triangle that form the right angle. They are also referred to as the adjacent and opposite sides in trigonometry, depending on the reference angle.In a triangle with sides 3, 4, and 5, the sides with lengths 3 and 4 are the legs. Trigonometric RatiosRatios that relate the angles of a right triangle to t...

Pythagorean Theorem a² + b² = c² In any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b). Use this to find a missing side when two sides are known. Trigonometric Ratios (SOH CAH TOA) sin(θ) = Opposite/Hypotenuse; cos(θ) = Adjacent/Hypotenuse; tan(θ) = Opposite/Adjacent Use these ratios to find a missing side when one side and one acute angle are known, or to find a missing angle when two sides are known (using the inverse functions). 45-45-90 Triangle Ratios Legs = x, Hypotenuse = x√2 In an isosceles right triangle (45-45-90), the two legs are congruent. The hypotenuse is √2 times the length of a leg. This allows for quick calculations without trigonometry. 30-60-90 Triangle...