8.10. Laws of Sines and Cosines http://www.ck12.org
10.m^6 A= 74 ◦,m^6 B= 11 ◦,BC= 16
11.m^6 A= 64 ◦,AB= 29 ,AC= 34
12.m^6 C= 133 ◦,m^6 B= 25 ◦,AB= 48Use the Law of Sines to solve 4 ABCbelow.
13.m^6 A= 20 ◦,AB= 12 ,BC= 5Recall that when we learned how to prove that triangles were congruent we determined that SSA (two sides and an
angle not included) did not determine a unique triangle. When we are using the Law of Sines to solve a triangle and
we are given two sides and the angle not included, we may have two possible triangles. Problem 14 illustrates this.
- Let’s say we have 4 ABCas we did in problem 13. In problem 13 you were given two sides and the not
included angle. This time, you have two angles and the side between them (ASA). Solve the triangle given
thatm^6 A= 20 ◦,m^6 C= 125 ◦,AC= 8. 4 - Does the triangle that you found in problem 14 meet the requirements of the given information in problem
13? How are the two differentm^6 Crelated? Draw the two possible triangles overlapping to visualize this
relationship.
Summary
This chapter begins with the Pythagorean Theorem, its converse, and Pythagorean triples. Applications of the
Pythagorean Theorem are explored including finding heights of isosceles triangles, proving the distance formula,
and determining whether a triangle is right, acute, or obtuse. The chapter then branches out into special right
triangles, 45-45-90 and 30-60-90. Trigonometric ratios, trigonometry word problems, inverse trigonometric ratios,
and the Law of Sines and Law of Cosines are explored at the end of this chapter.
Chapter Keywords
- Pythagorean Theorem
- Pythagorean Triple
- Distance Formula
- Pythagorean Theorem Converse
- Geometric Mean
- 45-45-90 Corollary
- 30-60-90 Corollary
- Trigonometry
- Adjacent (Leg)
- Opposite (Leg)
- Sine Ratio
- Cosine Ratio
- Tangent Ratio
- Angle of Depression
- Angle of Elevation
- Inverse Tangent
- Inverse Sine
- Inverse Cosine
- Law of Sines
- Law of Cosines