Find trigonometric ratios using right triangles

Learning Objectives Identify the hypotenuse, opposite, and adjacent sides relative to a specified acute angle in a right triangle. Define the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) using the acronym SOH CAH TOA and their reciprocal relationships. Calculate the exact values of all six trigonometric ratios for an acute angle in a right triangle when all three side lengths are known. Apply the Pythagorean theorem to determine a missing side length before calculating trigonometric ratios. Simplify trigonometric ratios, including those with radicals in the denominator, by rationalizing. Understand that the trigonometric ratios for a given angle are constant, regardless of the size of the right triangle, due to the properties of similar triangl...

TermDefinitionExample Right TriangleA triangle containing one angle that measures exactly 90 degrees.A triangle with side lengths 5, 12, and 13 is a right triangle because 5² + 12² = 13² (25 + 144 = 169). HypotenuseThe side of a right triangle that is opposite the 90-degree angle. It is always the longest side.In a 5-12-13 right triangle, the side with length 13 is the hypotenuse. Reference Angle (θ)The specific acute angle in a right triangle from which the 'opposite' and 'adjacent' sides are determined.In a triangle ABC with a right angle at C, if we are finding sin(A), then angle A is our reference angle. Opposite SideThe side across from the reference angle (θ).If our reference angle is A, the side that does not touch vertex A is the opposite side. Adjacent SideThe...

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 your reference angle θ. Reciprocal Trigonometric Ratios csc(θ) = 1/sin(θ) = Hypotenuse/Opposite sec(θ) = 1/cos(θ) = Hypotenuse/Adjacent cot(θ) = 1/tan(θ) = Adjacent/Opposite These three ratios are the multiplicative inverses of the primary ratios. They are used extensively in trigonometry and calculus for simplifying expressions and solving equations. Pythagorean Theorem a² + b² = c² In any right triangle where 'a' and 'b' are the lengths of the legs and 'c' is...