Introduction
SOH CAH TOA only works for right triangles. But the Law of Sines works for any triangle—it's our first tool for solving oblique (non-right) triangles.
Prerequisite Connection
You know SOH CAH TOA for right triangles. Now we're extending to all triangles.
Today's Increment
We're learning the Law of Sines and applying it to ASA and AAS triangles.
Why This Matters
Surveyors, pilots, and engineers use these techniques to calculate distances that can't be measured directly.
The Law of Sines
The Law of Sines
When to Use It
- • ASA: Two angles and the included side
- • AAS: Two angles and a non-included side
- • SSA: Two sides and a non-included angle (See Lesson 12.2)
Key Insight
Each ratio equals the diameter of the circumscribed circle! This is why all three ratios are equal.
Worked Examples
Example 1: AAS Triangle
In triangle ABC, , , and . Find side .
Set Up the Proportion
Solve for b
Calculate
Answer:
Example 2: ASA Triangle — Find Two Sides
In triangle ABC, , , and (the side between A and C). Find sides and .
Find Angle B
Find Side a
Find Side c
Answer:
Example 3: Complete Triangle Solution (Advanced)
In triangle ABC, , , and . Find all missing parts.
Find Angle B
Find Side a
Find Side c
Answer:
Common Pitfalls
Matching sides with wrong angles
Side is always opposite angle . Use consistent labeling!
Calculator in wrong mode
Check that your calculator is in DEGREE mode, not RADIAN mode.
Real-World Application
Surveying & Navigation
Surveyors use the Law of Sines to calculate distances across rivers, canyons, or other obstacles. By measuring angles from two known points and the distance between them, they can determine any other distance in the triangle without physically crossing the gap.
Practice Quiz
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