Honors Geometry Unit 7-B review By Narayan Prabhakar and Luke Horsburgh

1.Trigonometry (Solving for sides) (SOH CAH TOA) 2.Inverse Trigonometry (Solving for angles) (Sin -1 Cos -1 Tan -1 ) 3.Law of Sine - SinA/a = SinB/b = SinC/c 4.Law of Cosine - a 2 = b 2 + c 2 −2bc (CosA) 5.Solve right triangles using trigonometry and inverse trigonometry 6.Solve all kinds of triangles using Law of Sine and and Law of Cosine Overview of Key Concepts

Three Example Problems #1. (Easy) Solve the triangle #2. (Medium) Solve the triangle 14 Y X 22° 90°X° 14 X 33° 47° Y 90° 23° 40 X° X Y #3. (Hard) Solve the triangle X° *Remember that, due to the Triangle Sum Theorem, all angles in a triangle add up to 180°. That can save time on a test!*

Three Example Problems Answer Key #1. (Easy) Solve the triangle #2. (Medium) Solve the triangle ° 90°68° ° 47° ° 23° 40 67° #3. (Hard) Solve the triangle 100°

Connections to Other Concepts/Units ●This unit is strongly connected to the use of reference angles. Since sin is always positive in the second quadrant, there is possibly two triangles in an SSA triangle ●This unit is also connected to the pythagorean theorem. Many trigonometric ratios are founded from pythagorean triples. These include (1,√3,2) which is the triangle, and (1,1,√2) which is a triangle ●This can also be connected to graphing sin and cos functions. Since the amplitude and period of these functions is constant, the sin and cos functions repeat themselves. This is also why all trigonometric functions of coterminal angles are congruent. Coterminal means you can add or subtract 360 degrees to that angle

Common Mistakes/Struggles ●People sometimes switch around asa and aas triangles. This could lead to different solutions ●For SSA triangles, people may not necessarily look to see if there are 0,1, or 2 solutions. This could end up with people missing a possible triangle, and people coming up with imaginary triangles. ●When using the Law of Cosines, you have to use cosine. If you end up hitting the sin button, you will end up with a completely different answer because the only time when cosine and sine are equal are when it is with the angle 45 or any angle coterminal with 45. The Law of Sines is pretty easy but you have to know the cosine formula exactly. ●It is C 2 = A 2 +B 2 -2ABCosC

Real Life Example ●There are many different scenarios where trig can be used in the world ●But, let us take only one example ●Let’s say you are serving in tennis for practice. You want to hit the left most target you have placed on the service line which is a 25 foot diagonal away. You hit the ball when it is ten feet in the air. What is the angle in which you hit the ball a=10 and c=25, and you want to find the angle opposite to side a known as angle A.Sin A=.4 or 10/25. If you take the sin inverse of.4, you will get the angle you need which is approximately 26 degrees.

Real Life Example part B Now, let’s say that you want to find the horizontal distance to the target, otherwise known as side B. Since angle A is approximately 26 degrees, B= approximately 64 degrees. This brings us to another concept of solving triangles. By the law of sines, sin 26/10=sin64/x. If you multiply sin 64 and 10 and divide by sin 26 you get the answer to be about You can also verify this with the pythagorean theorem. Note: We didn’t use the angles 26 and 64, so we could make the answer as accurate as possible. We rounded to the hundredths digit.