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Chapter 9 Right Triangle Trigonometry Goals 9.1 Similar Right Triangles and Geometric Mean I will be able to identify the geometric mean of a right triangle when an altitude is drawn from vertex to hypotenuse. Using the geometric mean, I can set up a proportion comparing sides of the similar triangles formed by the altitude drawn. I can then solve my proportion correctly. 9.2 The Pythagorean Theorem Use the Pythagorean Theorem correctly to solve for a missing side of a right triangle, if given the other two side lengths. Prove the Pythagorean Theorem. Identify Pythagorean triples. 9.3 Converse of the Pythagorean Theorem Classify triangles as acute, right, or obtuse by using the Pythagorean Theorem when given the three side lengths of a triangle. 9.4 Special Right Triangles Use special triangle pattern for a 45-45-90 isosceles triangle to quickly find unknown side lengths. Use special triangle pattern for a 30-60-90 triangle to quickly find unknown side lengths. 9.5 Trigonometric Ratios Define sine, cosine, and tangent ratios. Correctly label the sides of a right triangle as opposite, adjacent, and hypotenuse given one acute angle. Set up a trigonometric ratio (sine, cosine, tangent) using two sides of a right triangle and a given acute angle measure. Set up a trig equation and solve for a missing side length given one acute angle of a right triangle, and one side length. Use the relationship between sine and cosine to compare sides and angles of triangles. 9.6 Solving Right Triangles Solve for a missing acute angle in a right triangle given two of the side lengths. Use inverse trig functions to solve the equation for a missing angle measure. Use the appropriate trig function (regular or inverse) to solve a right triangle completely. Chapter 9 Vocabulary 9.1 similar triangles altitude geometric mean hypotenuse 9.2 Pythagorean Theorem Pythagorean Triple 9.3 converse 9.4 short leg long leg hypotenuse base angles 9.5 trigonometric ratio sine (sin) cosine (cos) tangent (tan) angle of elevation angle of depression 9.6 solving a right triangle arcsine (sin-1) arccosine (cos-1) arctangent (tan-1) Simplifying Square Roots To simplify a square root, you “take out” anything that is a perfect square. Example: √63 What is the biggest perfect square that can be divided out of 63? √63 = √9 ∙ √7 = 3 ∙ √7 = 3√7 Let’s practice a few more. 1. √18 2. √72 3. √112 4. √132 5. √14 √6 “Three radical seven” 9.1 Notes: Geometric Mean & Similar Right Triangles Explore: Start with a right triangle, ∆𝐴𝐵𝐶, then draw an altitude from the right angle to the opposite side. Label the new point of intersection on the hypotenuse D. You have now created 3 similar right triangles!!! Recall that similar triangles have __________________angle measures and the sides are ______________________________________. If you separate, rotate, and line up the three triangles you can see the similarity. What segments are shown in more than one triangle? Use a highlighter to help you see. These segments that are “shared” between two triangles are called geometric means. When you set up proportions relating the sides of a triangle, they will be in the “means” position of your proportion. See below. Two small triangles: geo. mean is ̅̅̅̅ 𝐶𝐷 = ℎ𝑒𝑖𝑔ℎ𝑡 𝑏𝑎𝑠𝑒 = Two bigger triangles: geo. mean is ̅̅̅̅ 𝐴𝐶 = ℎ𝑒𝑖𝑔ℎ𝑡 𝑏𝑎𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = ℎ𝑒𝑖𝑔ℎ𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 Small & Big Triangle: ̅̅̅̅ geo. mean is 𝐵𝐶 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝑏𝑎𝑠𝑒 𝑏𝑎𝑠𝑒 Practice: First, label the angles of your triangles as “right”, “one”, and “two” Then, find the x and use the marked angles as a guide to set up your proportion. 1. 2. 3. Find TU 4. 5. 6. 6. What is the geometric mean of 18 and 2? Set up a proportion. Remember the “mean” is x. 9.2 Notes: The Pythagorean Theorem Pythagorean Theorem – USED FOR RIGHT TRIANGLES ONLY!!! If we have a right triangle, then 𝑎2 + 𝑏 2 = 𝑐 2 , is used to find the third side of a triangle if we are given two known side lengths. a, b represent the two legs of the triangle, they form the right angle c represents the hypotenuse of the triangle, across from the right angle (always longest) A ________________________________ ________________________ is a set of three positive integers, a,b, and c that satisfy the equation 𝑎2 + 𝑏 2 = 𝑐 2 . For example: Examples: 1. Try on your own: 4. 2. 3. 5. 6. Find the area of the triangle. First, find the height. Then, find the area of the triangle using base and height. 9.3 The Converse of the Pythagorean Theorem We know the Pythagorean Theorem, but how can I tell if I have a right triangle? Converse of Pythagorean Theorem If 𝑎2 + 𝑏 2 = 𝑐 2 , then we have a right triangle. The converse of the Pythagorean Theorem is used to determine if a triangle is right, obtuse or acute given 3 side lengths. Practice: Is this a right triangle? If not, is it obtuse or acute? FOLLOW THE HYPOTENUSE!!!! IS IT TOO SMALL? TOO BIG? OR JUST RIGHT? 1. 2. Practice: Can the following lengths represent the 3 sides of a triangle? If yes, what kind of triangle can be formed? 3. √250, 30, 34 4. ha ha ! 8, 13, 22 9.4 Notes: Special Right Triangles The 45-45-90 Triangle Hypotenuse = leg ∙ √𝟐 To find the LEGS – divide by √𝟐 To find the HYPOTENUSE – multiply by √𝟐 1. 2. GI = ____ HG = ____ 3. 4. 5. There is a square window with a diagonal length of 36√2 inches. What is the area of the window? The 30-60-90 Triangle hypotenuse = short leg ∙ 𝟐 Long leg = short leg ∙ √𝟑 ALWAYS FIND THE SHORT LEG FIRST!!! 5. 6. 7. 8. 9. 9.5 Notes: Trigonometric Ratios I. Introduction Every right triangle has two _______________ angles. From each of these acute angles, we can label the three sides of the triangle. Label the following triangle. Note: We never label from the _______________ angle. If you change your angle, your labels will change! Three basic trigonometric ratios (Memorize!!) Sine = Cosine = Tangent = SOH CAH TOA Let’s practice setting up the ratios and simplifying the fractions. 1. 2. 3. sin R = sin A = sin X = cos R = tan A = cos X = tan R = cos B = sin Y = tan S = tan B = cos Y = II. Relationships between Sine & Cosine. (DEGREE MODE!!!) 1. sin 40 = 2. cos 36 = cos 50 = sin 54 = The sine & cosine of ___________________________________ angles are always _______________________! Complementary angles add to __________. Therefore, we can define them in terms of the other. 𝑚∠𝐴 = 𝑚∠𝐷 = 𝑚∠𝐶 = 𝑚∠𝐹 = 𝑠𝑖𝑛 𝐴 = cos 𝐷 = sin 𝐶= cos 𝐹 = _________________________________________________________________________________________________ Practice Problems: 1. An equation is shown. sin(𝑎°) = cos(𝑏°) 0 < 𝑎 < 90 and 0 < 𝑏 < 90 Write an expression for b in terms of a. 2. b = __________________________ III. Solving for missing side lengths using sin-cos-tan Setting up equations: First, always label triangle from your given angle. o Opposite, adjacent, hypotenuse. Then ask yourself: o “What side was I given?” o “What side am I trying to find?” Identify which ratios to use from these questions! Set up correct equation. (Remember to include your angle measure) Solve equation using normal inverse operations and answer the question. Practice. 1. Solve for x 2. Find the height of the pole. 3. Find the hypotenuse 4. Solve for s and r 5. Solve for the missing length, x. 9.6 Notes: Solving Right Triangles I. Solving for Missing Angles using Sin-Cos-Tan Inverse Trigonometric Functions The inverse of sin A is __________________________ or ________________. The inverse of cos A is _________________________ or ________________. The inverse of tan A is __________________________ or _______________. To solve or missing angles using the inverse trigonometry functions: 1. Set up equations the same as before but notice your angle will be a variable. 2. Solve using inverse operations. Practice finding both angle measures. a. b. II. Solving Right Triangles When we solve a right triangle, we find all three _________________________________________ and all three _____________________________________________. Let’s solve a few. (Round decimals to the nearest tenth) 1. 2. Solving Right Triangles Chart I HAVE I WANT I WILL USE 2 Sides 3rd Side Pythagorean Theorem (9.2) 1 side of 45-45-90 triangle Other two sides 1 side of 30-60-90 triangle Other two sides (9.4) (9.4) 1 side and 1 angle (other than 90 degree angle) 2 sides Another side Sin-Cos-Tan (9.5) An angle Inverse (or arc) Sin-Cos-Tan (9.6) Remember this is only for right triangles! If you don’t have a right angle, these methods won’t work! Law of Sines & Law of Cosines Notes (extension of chapter 9 Trigonometry) The trigonometric ratios we previously learned (sine, cosine, and tangent) can only be used to solve right triangles. Today we will learn to solve oblique triangles. You can use the Law of Sines to solve triangles when two angles and the length of any side are known (AAS or ASA cases), or when the lengths of two sides and an angle opposite one of the two sides are known (SSA case – be careful for the ambiguous case!). You must have one opposite side and angle to create a full ratio! LAW OF SINES ratios. Use two together to create and solve proportions. 𝑎 𝑏 𝑐 = = sin 𝐴 sin 𝐵 sin 𝐶 OR sin 𝐴 sin 𝐵 sin 𝐶 = = 𝑎 𝑏 𝑐 I. Finding sides/angles in oblique triangles using law of sines. 1. Example using SSA case: Solve the triangle for side c, angle B, and angle C. 2. Example using AAS case: Solve the triangle for side a, side c, and angle A 3. Example using ASA case: Solve the triangle for side a, side c, and angle C. II. Finding the area of an oblique triangle using the sine ratio. How do we get the height? Write an expression for height in terms of side c and angle A. Write an area formula in terms of sides b, c, and angle A. Oblique area formulas: 1 𝐴 = 𝑏𝑐 ∙ sin 𝐴 2 𝐴= 1 2 𝑎𝑐 ∙ sin 𝐵 4. Find the area of the triangle. Round to nearest tenth. 𝐴= 1 2 𝑎𝑏 ∙ sin 𝐶 You can use the Law of Cosines to solve triangles when two sides and the included angle are known (SAS case), or when all three sides are known (SSS case). Law of Cosines 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 ∙ 𝑐𝑜𝑠 𝐴 𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 ∙ 𝑐𝑜𝑠 𝐵 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 ∙ cos 𝐶 III. Finding sides/angles in oblique triangles using law of cosines. 5. Example using SAS case: Solve the triangle by finding side b, angle A and angle C. 6. Example using SSS case: Solve the triangle by finding angle A, angle B, and angle C.