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- Sine Rule and Cosine Rule
Law of sines
- Intro Lesson12:58
- Lesson: 1a4:10
- Lesson: 1b4:01
- Lesson: 24:28
- Lesson: 315:18
Law of sines
In this section, we will learn about the Law of Sines, also known as the Sines Rule. The Law of Sines is a formula that models the relationship between the sides and the angles of any triangle, be it a right-angled triangle, an obtuse triangle, or an acute triangle. In order to use the Law of Sines, we need to satisfy the "one pair, one additional information" condition (i.e. Angle-Angle-Side abbreviated as AAS, and Angle-Side-Angle abbreviated as ASA). We will also explore the concept of the Ambiguous Case of the Law of Sines.
Basic Concepts: Use sine ratio to calculate angles and sides (Sin = ho ), Use cosine ratio to calculate angles and sides (Cos = ha ), Use tangent ratio to calculate angles and sides (Tan = ao )
Related Concepts: Quotient identities and reciprocal identities, Pythagorean identities, Sum and difference identities
Lessons
Law of Sine
For any △ ABC,
sin(A)a =sin(B)b =sin(C)c
and,
asin(A) =bsin(B) =csin(C)
Use the Law of Sine when given a pair!
Ambiguous case
Ambiguous case of the Law of Sine arises when given SSA (side-side-angle)
Step 1) Use the given angle to find the height of the triangle: h=bsin(A)
Step 2) Check if,
Sidea < h, then no triangles
Sidea=h, then 1 triangle
Sidea > h, then 1 triangle
h < Sidea < Sideb, then 2 triangles
Step 3) Solve the triangle(s)!
For any △ ABC,
sin(A)a =sin(B)b =sin(C)c
and,
asin(A) =bsin(B) =csin(C)
Use the Law of Sine when given a pair!
Ambiguous case
Ambiguous case of the Law of Sine arises when given SSA (side-side-angle)
Step 1) Use the given angle to find the height of the triangle: h=bsin(A)
Step 2) Check if,
Sidea < h, then no triangles
Sidea=h, then 1 triangle
Sidea > h, then 1 triangle
h < Sidea < Sideb, then 2 triangles
Step 3) Solve the triangle(s)!
- Introduction
- 1.Given the following triangle △ABC,
a)Solve for∠Cb)Solve for a - 2.Solve for side x
- 3.Ambiguous case: SSA triangles
In △DEF, DE=21cm, ∠F=45°, and EF=24cm; find DF.