Trigonometric ratios in similar right triangles

Learning Objectives Define the three primary trigonometric ratios: sine, cosine, and tangent. Identify corresponding angles and sides in similar right triangles. Prove, using similarity, that the value of a trigonometric ratio depends only on the measure of the angle, not the size of the triangle. Calculate the sine, cosine, and tangent of an acute angle in a right triangle given its side lengths. Use the properties of similar right triangles to determine the trigonometric ratios for a given angle. Solve for unknown side lengths in a right triangle using trigonometric ratios derived from a similar triangle. Ever wondered how video game designers can make a character appear to get closer or farther away without distorting their shape? 🎮 It's all about scaling, which is...

TermDefinitionExample Similar TrianglesTriangles that have the same shape but possibly different sizes. Their corresponding angles are equal, and the ratios of their corresponding side lengths are constant.A triangle with sides 3, 4, 5 is similar to a triangle with sides 6, 8, 10 because all sides of the second triangle are twice the length of the sides of the first, and their angles are identical. Right TriangleA triangle that has one angle measuring exactly 90 degrees.A triangle with angles 30°, 60°, and 90°. HypotenuseThe side of a right triangle that is opposite the 90-degree angle. It is always the longest side.In a right triangle with sides 5, 12, and 13, the side with length 13 is the hypotenuse. Opposite SideThe side across from a specific reference angle (that is not the right an...

Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. This is the primary way to prove two right triangles are similar. Since both triangles already have a 90-degree angle, you only need to show that one other corresponding acute angle is congruent. Primary Trigonometric Ratios (SOH CAH TOA) For a given acute angle θ in a right triangle: Sine: sin(θ) = Opposite / Hypotenuse Cosine: cos(θ) = Adjacent / Hypotenuse Tangent: tan(θ) = Opposite / Adjacent These three formulas define the fundamental trigonometric ratios. The acronym SOH CAH TOA is a mnemonic device to remember them. The key insight of this lesson is that for any two similar right triangles, the value of sin(θ), cos(θ),...