9.6 Solving Right Triangles Geometry EQ: How do you use trig ratios to find missing acute angles of a right triangles when you know the side lengths?

Geometry 9.6 Solving Right Triangles Goals Use inverse trig functions to find angle measures. Solve right triangles. Solve problems using right triangles. May 16, 2018 Geometry 9.6 Solving Right Triangles

Solving a triangle means… Finding the lengths of the three sides. Finding the measure of the three angles. In a right triangle, one angle is always 90 and we don’t need to worry about it. A c b B a C May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles We can use… Trig equations Pythagorean Theorem Inverse trig functions May 16, 2018 Geometry 9.6 Solving Right Triangles

Inverse Trig Functions If sin A = x, then sin-1x = A. If cos A = x, then cos-1x = A. If tan A = x, then tan-1x = A. May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Example 1 Sin A = 0.7660. What is A? Sin-1(.766) = A Look at your trig table Under the column for Sine, find the closest value to .766 A  50 Without a calculator, getting any more accurate results is exceedingly difficult. May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Example 2 Use a trig table or inverse function on calculator. Cos A = 0.2079. What is A? Cos-1(.2079) = A A  78 Be careful, remember angle measures wont be negative and the ratio to side lengths and angle measures need to make sense. Its easy to make errors working too fast or carelessly on calculators. May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Example 3 Tan A = 0.1051. What is A? Tan-1(.1051) = A A  6 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Solving a triangle First, we will find A. tan A = 7/12 tan A  .5833 tan-1(.5833) = A The closest we find is .5774 under tangent. A  30 A c 12 B 7 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Solving a triangle Now find B. Since A and B are complementary, B is about 60. A 30 c 12 B 7 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Solving a triangle Find side c. Pythagorean Theorem A 30 c 12 60 B 7 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Solving a triangle The triangle is solved. Notice: the measures are all approximate. A 30 13.9 12 60 B 7 May 16, 2018 Geometry 9.6 Solving Right Triangles

You try it. Solve the triangle. First, find angle A. tan A = 32/15 tan A  2.1333 tan-1(2.1333) = A A  65 A c 15 B 32 May 16, 2018 Geometry 9.6 Solving Right Triangles

You try it. Solve the triangle. Next, find angle B. 90 – 65 = 25 A 65 c 15 B 32 May 16, 2018 Geometry 9.6 Solving Right Triangles

You try it. Solve the triangle. Now find side c. A 65 c 15 25 B 32 May 16, 2018 Geometry 9.6 Solving Right Triangles

You try it. Solve the triangle. The triangle is solved. A 65 35.3 15 25 B 32 May 16, 2018 Geometry 9.6 Solving Right Triangles

Example Solve the triangle. Find A first, since it’s the complement of the other acute angle. A = 90 – 38 = 52 52 16.5 b 38 a May 16, 2018 Geometry 9.6 Solving Right Triangles

Example Solve the triangle. Now use sine to find a. 52 16.5 b 38 a May 16, 2018 Geometry 9.6 Solving Right Triangles

Example Solve the triangle. Now use cosine to find b. 52 16.5 b 38 13.0 May 16, 2018 Geometry 9.6 Solving Right Triangles

Example Solve the triangle. The triangle is solved. 52 16.5 10.2 38 13.0 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles 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 not to use rounded data in any calculation. Be very careful using a calculator. Check everything twice. May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Solve this triangle. A c 25 B 10 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Solution A c2 = 252 + 102 c2 = 725 c  26.9 tan B = 25/10 B = tan-1(2.5) B = 68 A = 90 – 68 = 22 c 26.9 25 22 68 B 10 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Indirect Measure One of the most powerful uses of trig is to measure things that can’t be measured directly. This is indirect measure. Fundamental process used in surveying, map making, astronomy and other applications. May 16, 2018 Geometry 9.6 Solving Right Triangles

Problem 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. How far is he from each object? 110 ft. Jim 7 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Problem Solution 110 ft. Jim 7 x May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Problem Solution y 110 ft. Jim 7 896 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Is this correct? YES! 902 110 ft. Jim 7 896 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles Indirect Measure Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly. 902 110 ft. Jim 7 896 May 16, 2018 Geometry 9.6 Solving Right Triangles

Geometry 9.6 Solving Right Triangles 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 given values when possible. May 16, 2018 Geometry 9.6 Solving Right Triangles