- Home
- Trigonometry
- Trigonometric Ratios and Angle Measure
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.
Do better in math today
2.
Trigonometric Ratios and Angle Measure
2.1
Angle in standard position
2.2
Coterminal angles
2.3
Reference angle
2.4
Find the exact value of trigonometric ratios
2.5
ASTC rule in trigonometry (All Students Take Calculus)
2.6
Unit circle
2.7
Converting between degrees and radians
2.8
Trigonometric ratios of angles in radians
2.9
Radian measure and arc length
2.10
Law of sines
2.11
Law of cosines
2.12
Applications of the sine law and cosine law