Find trigonometric ratios using right triangles

Learning Objectives Define the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) in the context of a right triangle. Correctly identify the opposite, adjacent, and hypotenuse sides relative to a given acute angle in any right triangle. Calculate the exact values of the six trigonometric ratios for an acute angle when all three side lengths of a right triangle are known. Apply the Pythagorean theorem to determine a missing side length before calculating the trigonometric ratios. Determine the trigonometric ratios of a complementary angle, given the ratios of the other acute angle. Simplify basic trigonometric expressions by substituting their right-triangle ratio definitions. How do surveyors map out land or engineers design a bridge without measur...

TermDefinitionExample Right TriangleA triangle containing one angle that measures exactly 90 degrees.A triangle with side lengths 3, 4, and 5 is a right triangle because 3² + 4² = 5². HypotenuseThe longest side of a right triangle. It is always the side directly opposite the 90-degree angle.In a 3-4-5 right triangle, the side with length 5 is the hypotenuse. Opposite SideThe side across from a given reference angle (θ) in a right triangle.In a 3-4-5 triangle, if θ is the angle between sides 4 and 5, the opposite side is the side with length 3. Adjacent SideThe side next to a given reference angle (θ) that is not the hypotenuse.In a 3-4-5 triangle, if θ is the angle between sides 4 and 5, the adjacent side is the side with length 4. Trigonometric RatioA ratio comparing the lengths of two s...

Primary Trigonometric Ratios (SOH CAH TOA) sin(θ) = Opposite/Hypotenuse cos(θ) = Adjacent/Hypotenuse tan(θ) = Opposite/Adjacent This mnemonic is the foundation for defining the three primary trigonometric functions. Use it to set up the correct ratio based on the sides relative to the angle θ. Reciprocal Trigonometric Ratios csc(θ) = 1/sin(θ) = H/O sec(θ) = 1/cos(θ) = H/A cot(θ) = 1/tan(θ) = A/O These rules define the remaining three trigonometric functions. They are calculated by taking the reciprocal of the primary ratios. Pythagorean Theorem a² + b² = c² In a right triangle where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse, this theorem is used to find the length of an unknown side when two sides ar...