8-8 Special Right Triangles and Trigonometry notes

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent Triangles8-8 Special Right Triangles and TrigonometryWarm-up1. Find the length of the hypotenuse in ΔABC. AC ≈ 15.82. In ΔRWT, what side is opposite

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesIn a 30º-60º-90º triangle, the measure of the hypotenuse is twice the measureof the shorter leg. The measure of the longer leg is √3 times the measure ofthe shorter leg.Notice: the shorter leg is the leg connecting the 60°and 90° angles.Example 1Find the approximate measure of each unknown side in each triangle.a.SolutionXY =a√2 = 2 and XZ = ZY = aso XY = XZ(√2) now solve for XZ2 = XZ(√2)2= 1.4 so XZ = ZY = 1.42b. SolutionJK = 12 ← Hypotenuse (2b)JL ← shorter leg ( b), so 1 • JK = 1 • 12 = 62 2KL ← longer leg (b√3), so ( JL ) 3 = 6 3 ≈10.48-8 Special Right Triangles and Trigonometry

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesExample 2Find the measure of the unknown sides.a. b. c.Solutiona. 45° - 45° - 90° Δ because it is an isosceles right triangle.SR ← leg = a = 10TR ← leg = a = 10ST ← hypotenuse = a√2ST = 10√2≈ 14.14b. 45° - 45° - 90° ΔPN ← hypotenuse =a√2 = 9√2So, a = 9PM ← leg = a = 9NM ← leg = a = 9c. 30° - 60° - 90° ΔJL ← shorter leg = b = 5KL ← longer leg = b√3 = 5√3KJ ← hypotenuse = 2b = 10Trigonometric Ratiosmeasureof leg opposite∠ABCsin A ==measureof hypotenuseABmeasure of leg adjacent to ∠ A ACcos A = =measure of hypotenuse ABmeasure of leg opposite ∠ A BCtan A = =measure o f leg adjacent to ∠AACBlegopposite‹AChypotenuseleg adjacent to ‹AA8-8 Special Right Triangles and Trigonometry

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesSide-noteThree ways to solve for an unknown side.• Special right triangle relationships (30-60-90, 45-45-90 triangles)• Pythagorean theorem (unless only one side is known)• Trig Ratios (sin, cos, tan)“SOHCAHTOA”Trigonometric ratios for special right triangles30º angle45º angle60º anglesin =oppositehypotenuseb 1=2b 2a 1=a 2 2b 3 3=2b 2cos =adjacenthypotenuseb 3 3=2b 2a 1=a 2 2b 1=2b 2tan = oppositeadjacentb 1=b 3 3a =1ab 3 = 3bSide noteDon’t rely on the table. Be able to use Pythagorean Theorem, trig ratios, and the specialright angle relations to find angle and/or side measures.Example 3Find the value of each variable.a. sin xº =32Think, which Δ contains √3? 30° - 60° - 90° Δ. So, x = 60°b. cos yº =32Think, which Δ contains √3? 30° - 60° - 90° Δ. So, y = 30°c. tan zº = 1 z = 45°The value of a trig ratio depends only on the measure of the angle, not on the size of thetriangle.8-8 Special Right Triangles and Trigonometry

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesExample 4: Find the value of the variable.a. b.Solution 4a. 30° - 60° - 90° Δ b. 45° - 45° - 90° Δ

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesExample 7Find the approximate measure of each unknown side in the triangle.a. YXY = 2(4.5) = 9ZX = 4.5√3 ≈ 7.860º4.5Z30ºX8-8 Special Right Triangles and Trigonometry