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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.
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13.
Trigonometry
13.1
Angle in standard position
13.2
Coterminal angles
13.3
Reference angle
13.4
Find the exact value of trigonometric ratios
13.5
ASTC rule in trigonometry (All Students Take Calculus)
13.6
Unit circle
13.7
Converting between degrees and radians
13.8
Trigonometric ratios of angles in radians
13.9
Radian measure and arc length
13.10
Law of sines
13.11
Law of cosines
13.12
Applications of the sine law and cosine law