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9.6 Warmup Find the value of x. Then find the ratio of sin θ , cos θ , and tan θ for the triangle. 12 1. 2. 3. 12 θ 5 θ 6 Find the value of the unknown sides. 4. 5. April 1, 2016 Geometry 9.6 Solving Right Triangles 6. ? Classify. 7, 9, 12 1 Geometry 9.6 Solving Right Triangles 9.6 Essential Question When you know the lengths of the sides of a right triangle, how can you find the measures of the two acute angles? April 1, 2016 Geometry 9.6 Solving Right Triangles 3 Goals Use inverse trig functions to find angle measures. Solve right triangles. Solve problems using right triangles. April 1, 2016 Geometry 9.6 Solving Right Triangles 4 Solving a triangle means… Finding the lengths of the three sides. Finding the measure of the three angles. In a right A c B April 1, 2016 a Geometry 9.6 Solving Right Triangles b C triangle, one angle is always 90, thus we don’t need to worry about it. 5 Our Tools to Solve Triangles: Trig equations Pythagorean Theorem Inverse trig functions The sum of int. angles of a triangle A calculator (“Best Friend Calculator Guy”) – for speed and accuracy April 1, 2016 Geometry 9.6 Solving Right Triangles 6 Inverse Trig Function 𝑨 𝑖𝑠 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑥 𝑖𝑠 𝑎 𝑡𝑟𝑖𝑔 𝑟𝑎𝑡𝑖𝑜 Trig Function sin 𝐴 = 𝑥 means: the sine of an angle is a trig ratio. Inverse Trig Function sin−1 𝑥 = 𝐴 means: the inverse sine of a trig ratio is an angle. Example: sin 80 = 0.9848 sin-1 .9848 = 80 April 1, 2016 Geometry 9.6 Solving Right Triangles 7 Inverse Trig Functions If sin A = x, then sin-1x = A. 𝑜 −1 sin ℎ =𝐴 If cos A = x, then cos-1x = A. sin 𝐴 = 𝑜 ℎ 𝑐𝑜𝑠 𝐴 = 𝑎 ℎ If tan A = x, April 1, 2016 𝑡𝑎𝑛 𝐴 = 𝑜 𝑎 𝑎 −1 c𝑜𝑠 =𝐴 ℎ then tan-1x = A. −1 𝑜 tan 𝑎 =𝐴 Geometry 9.6 Solving Right Triangles 8 Example 1 sin A = 0.7660. What is A? A = sin-1(.766) A 50 April 1, 2016 Geometry 9.6 Solving Right Triangles 9 Example 2 cos A = 0.2079. What is A? A = cos-1(.2079) A 78 April 1, 2016 Geometry 9.6 Solving Right Triangles 10 Example 3 tan A = 0.1051. What is A? A = tan-1(.1051) A 6 April 1, 2016 Geometry 9.6 Solving Right Triangles 11 Example 4: Solving a triangle First, we will find A. A tan A = 7/12 A = tan-1(7/12) c 12 A 30.3 7 April 1, 2016 B Geometry 9.6 Solving Right Triangles 12 Example 4: Solving a triangle Now find B. A tan B = 12/7 12 30.3 B = tan-1(12/7) c A 59.7 7 April 1, 2016 B Geometry 9.6 Solving Right Triangles 13 Example 4: Solving a triangle Or… A 12 30.3 The acute angles of a right triangle are complementary. c B = 90 – 30.3 = 59.7 59.7 7 April 1, 2016 B Geometry 9.6 Solving Right Triangles 14 Example 4: Solving a triangle Find side c. A 12 30.3 Pythagorean Theorem is best because it doesn’t use rounded data. c c 12 7 2 April 1, 2016 2 c 144 49 2 59.7 7 2 B c 2 193 c 13.9 Geometry 9.6 Solving Right Triangles 15 Example 4: Solving a triangle The triangle is solved. A 12 Notice: the measures are all approximate. 30.3 13.9 59.7 7 April 1, 2016 B Geometry 9.6 Solving Right Triangles 16 You try it. Solve the triangle. First, find angle A. tan A = 32/15 A A = tan-1(32/15) c A 64.9 15 32 April 1, 2016 B Geometry 9.6 Solving Right Triangles 17 You try it. Solve the triangle. Next, find angle B. A tan B = 15/32 B = tan-1(15/32) 64.9 c B 25.1 15 32 April 1, 2016 B or… 90 – 64.9 = 25.1 Geometry 9.6 Solving Right Triangles 18 You try it. Solve the triangle. c 15 32 Now find side c. 2 A 2 2 c 225 1024 2 64.9 c c 1249 15 2 25.1 32 B c 1249 c 35.3 April 1, 2016 Geometry 9.6 Solving Right Triangles 19 You try it. Solve the triangle. The triangle is solved. A 64.9 35.3 15 25.1 32 April 1, 2016 B Geometry 9.6 Solving Right Triangles 20 Example 5: Solve the triangle. A 52 b 16.5 Find A first, since it’s the complement of the other acute angle. A = 90 – 38 = 52 38 a April 1, 2016 Geometry 9.6 Solving Right Triangles 21 Example 5: Solve the triangle. A Now use sine to find a. 52 b 16.5 38 a April 1, 2016 a sin 52 16.5 16.5sin 52 a a 13 Geometry 9.6 Solving Right Triangles 22 Example 5: Solve the triangle. A Now use cosine to find b. 52 b 16.5 38 13.0 April 1, 2016 b cos52 16.5 16.5cos52 b b 10.2 Geometry 9.6 Solving Right Triangles 23 Example 5: Solve the triangle. A The triangle is solved. 52 10.2 16.5 38 13.0 April 1, 2016 Geometry 9.6 Solving Right Triangles 24 Important You can solve a triangle in any order you want to, as long you have the data you need for each step. It’s best to not use rounded data in any calculation. Be very careful using a calculator. CHECK EVERYTHING TWICE!! April 1, 2016 Geometry 9.6 Solving Right Triangles 25 Your Turn: Solve this triangle. A 25 c 10 April 1, 2016 B Geometry 9.6 Solving Right Triangles 26 Your Turn: Solution A c2 = 252 + 102 c2 = 725 25 c 26.9 c26.9 tan B = 25/10 B = tan-1(25/10) 68.2 10 B B = 68.2 A = 90 – 68.2 = 21.8 April 1, 2016 Geometry 9.6 Solving Right Triangles 27 Indirect Measure One of the most powerful uses of trig is to measure things that can’t be measured directly. This is indirect measure. It’s a fundamental process used in surveying, map making, astronomy and other applications. April 1, 2016 Geometry 9.6 Solving Right Triangles 28 Example 6: Using a transit. Jim the Surveyor uses a transit to measure distances. He knows the distance between the tree and the fire hydrant is 110 ft. And to move from one to the other he swings his transit through 7.5. How far is he from each object? 110 ft. Jim 7.5 April 1, 2016 Geometry 9.6 Solving Right Triangles 29 Example 6: Solution 110 tan 7.5 x 110 x tan 7.5 x 835.5 110 ft. Jim 7.5 x April 1, 2016 Geometry 9.6 Solving Right Triangles 30 Example 6: Solution 110 sin 7.5 y 110 y sin 7.5 y 842.7 y Jim 110 ft. 7.5 835.5 April 1, 2016 Geometry 9.6 Solving Right Triangles 31 Example 6: Is this correct? YES! 110 ft. Jim 7.5 835.5 April 1, 2016 Geometry 9.6 Solving Right Triangles 32 Example 6: Indirect Measure Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly. 110 ft. Jim 7.5 835.5 April 1, 2016 Geometry 9.6 Solving Right Triangles 33 Summary Solving a triangle means to find all six parts: 3 angles, 3 sides. Use inverse trig function (sin-1, cos-1, tan-1) to find angles. Use the given data to calculate values, when possible. April 1, 2016 Geometry 9.6 Solving Right Triangles 34 Homework April 1, 2016 Geometry 9.6 Solving Right Triangles 35