Mathematics
Grade 10
15 min
Law of Sines
Law of Sines
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
State the Law of Sines formula from memory.
Identify the conditions required to use the Law of Sines (AAS, ASA, SSA).
Use the Law of Sines to calculate an unknown side length in an oblique triangle.
Use the Law of Sines to calculate an unknown angle measure in an oblique triangle.
Recognize the Ambiguous Case (SSA) and determine if zero, one, or two triangles are possible.
Solve real-world problems by applying the Law of Sines.
How can surveyors measure the distance across a wide river without ever crossing it? 🗺️ The Law of Sines gives us the tools to measure the unreachable!
This tutorial introduces the Law of Sines, a powerful formula used to find unknown side lengths and angle measures in any triangle, not just right-angled ones. You will learn when...
2
Key Concepts & Vocabulary
TermDefinitionExample
Oblique TriangleAny triangle that is not a right-angled triangle. It can be either an acute triangle (all angles are less than 90°) or an obtuse triangle (one angle is greater than 90°).A triangle with angles 70°, 60°, and 50° is an oblique (and acute) triangle.
Opposite Angle/Side PairIn any triangle, a side is 'opposite' the angle that is not one of its endpoints. We label angles with uppercase letters (A, B, C) and their opposite sides with corresponding lowercase letters (a, b, c).In triangle ABC, side 'a' is opposite angle A, side 'b' is opposite angle B, and side 'c' is opposite angle C.
Angle-Side-Angle (ASA)A case where we know the measures of two angles and the length of the side included between them.Given Angle A = 4...
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Core Formulas
The Law of Sines (for finding sides)
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
This formula states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. This form is most useful when you are trying to solve for an unknown side length.
The Law of Sines (for finding angles)
\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}
This is the reciprocal form of the Law of Sines. It is algebraically identical to the standard form but is more convenient to use when you need to solve for an unknown angle, as the unknown (sin(A), sin(B), or sin(C)) is in the numerator.
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Challenging
The Ambiguous Case (SSA) can result in two triangles because the sine function has the property that sin(θ) = sin(180° - θ). This means that for a calculated value of sin(B) = x, the angle B could be...
A.Only the acute angle θ = sin⁻¹(x).
B.Only the obtuse angle θ = 180° - sin⁻¹(x).
C.Either the acute angle θ or the obtuse angle 180° - θ, provided the latter forms a valid triangle.
D.Any angle θ + 360k for any integer k.
Challenging
Given triangle ABC with Angle A = 30° and side b = 12. For which range of values for side 'a' will there be exactly two possible triangles?
A.a < 6
B.a = 6
C.6 < a < 12
D.a > 12
Challenging
In triangle ABC, Angle B = 60° and side c = 10. What exact value must side 'b' have to form exactly one, right-angled triangle?
A.10
B.5
C.5√3
D.10√3
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