An oblique triangle is any triangle that is not a right triangle (no 90° angle).
For oblique triangles, we cannot use basic trigonometry (SOH CAH TOA). Instead, we use:
Law of Sines: Relates sides and their opposite angles
Law of Cosines: Generalizes the Pythagorean theorem
Triangle Notation
For any triangle with vertices A, B, and C:
Angles: , , (or simply , , )
📚 Practice Problems
1Problem 1easy
❓ Question:
In triangle ABC, A=35°, , and cm. Find the length of side .
Explain using:
⚠️ Common Mistakes: Law of Sines and Law of Cosines
Avoid these 4 frequent errors
🌍 Real-World Applications: Law of Sines and Law of Cosines
See how this math is used in the real world
📝 Worked Example: Related Rates — Expanding Circle
Problem:
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 10 cm?
Apply the Law of Sines and Law of Cosines to solve oblique triangles and find missing sides and angles.
How can I study Law of Sines and Law of Cosines effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
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Law of Sines and Law of Cosines is part of the AP Precalculus course on Study Mondo, specifically in the Trigonometric Functions section. You can explore the full course for more related topics and practice resources.
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∠A
∠B
∠C
A
B
C
Sides:
Side a is opposite angle A
Side b is opposite angle B
Side c is opposite angle C
Sum of angles:A+B+C=180° (or π radians)
Law of Sines
Formula
sinAa=sinBb=sinCc
Or equivalently:
asinA=bsinB=csinC
When to Use
Use the Law of Sines when you have:
AAS (Angle-Angle-Side): Two angles and one side
ASA (Angle-Side-Angle): Two angles and the included side
SSA (Side-Side-Angle): Two sides and a non-included angle ⚠️ Ambiguous case
Solving with Law of Sines
Given: Two angles and one side (AAS or ASA)
Steps:
Find the third angle: C=180°−A−B
Use the law of sines to find the unknown sides
Use the ratio sinAa=sinBb
The Ambiguous Case (SSA)
When given two sides and a non-included angle (SSA), there may be:
0 solutions (no triangle exists)
1 solution (one unique triangle)
2 solutions (two different triangles)
Why it's ambiguous: The side opposite the known angle might "swing" to create two different triangles.
To determine the number of solutions:
Given sides a, b and angle A (where a is opposite A):
If a<bsinA: No triangle (side too short)
If a=bsinA: One triangle (right triangle)
If bsinA<a<b: Two triangles (ambiguous case)
If a≥b: One triangle
Law of Cosines
Formulas
For any triangle:
a2=b2+c2−2bccosAb2=a2+c2−2accosBc2=a2+b2−2abcosC
Note: When C=90°, cosC=0, and this reduces to the Pythagorean theorem: c2=a2+b2
Solving for an Angle
Rearrange to solve for the cosine:
cosA=2bcb2+c2−a2cosB=2aca2+c2cosC=2aba2+b2−
Then use A=arccos(2bcb2+c2−a2)
When to Use
Use the Law of Cosines when you have:
SAS (Side-Angle-Side): Two sides and the included angle
SSS (Side-Side-Side): All three sides
Strategy for Solving Triangles
Given Information → Method
Given
Method
Steps
AAS or ASA
Law of Sines
Find third angle, then use ratios
SAS
Law of Cosines
Find third side, then use Law of Sines
SSS
Law of Cosines
Find one angle, then use Law of Sines
SSA
Law of Sines
Check ambiguous case first
Area of a Triangle
Using the Law of Sines, we can derive:
Area=21absinC=21bcsinA=21acsinB
This is useful when you know two sides and the included angle.
Common Applications
Navigation: Finding distances and bearings
Surveying: Measuring inaccessible distances
Engineering: Analyzing forces in structures
Physics: Resolving vector components
Tips for Success
Draw a diagram and label all known values
Identify the given information (AAS, SAS, SSS, etc.)
Choose the appropriate law (Sines or Cosines)
Check your answer using the angle sum (A+B+C=180°)
Watch for the ambiguous case with SSA
Use calculator in correct mode (degrees or radians)
B
=
65°
c=10
a
💡 Show Solution
Solution:
Given:
A=35°
B=65°
c=10 cm
Find:a
Step 1: Identify the case
This is AAS (two angles and one side), so use the Law of Sines.
Step 2: Find the third angleC=180°−A−B=180°−35°−65°=80°
Step 3: Apply Law of SinessinAa=sinC
Substitute:
sin35°a=sin80°
Step 4: Solve for aa=sin80°10sin35°
Calculate:
a=0.984810×0.5736≈0.9848
Answer:a≈5.82 cm
Verification:
All angles sum to 180° ✓
a<c makes sense since A<C (smaller angle opposite smaller side) ✓
2Problem 2medium
❓ Question:
In triangle ABC, a=12, b=15, and A=35°.
a) Use the Law of Sines to find angle B.
b) Find angle C.
c) Find side c.
💡 Show Solution
Solution:
Part (a): Law of Sines: sinAa=
3Problem 3medium
❓ Question:
In triangle ABC, a=8, b=5, and C=60°. Find the length of side c.
💡 Show Solution
Solution:
Given:
a=8
b=5
C=
4Problem 4medium
❓ Question:
In triangle ABC, a=8, b=11, and c=14.
a) Use the Law of Cosines to find angle C.
b) Find the area of the triangle.
💡 Show Solution
Solution:
Part (a): Law of Cosines: c2=a2+b
5Problem 5hard
❓ Question:
In triangle ABC, a=20, b=15, and c=25. Find all three angles.
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.
−
b2
c2
c
10
5.736
≈
5.82 cm
sinBb
sin35°12=sinB15
sinB=1215sin35°=1215(0.5736)=128.604=0.717
B=sin−1(0.717)≈45.8°
Note: There could be another solution B′=180°−45.8°=134.2°, but we need to check if it's valid.
If B=134.2°, then A+B=35°+134.2°=169.2°<180°, so this is also possible.
However, since b>a and angle B is opposite the larger side, we expect B>A.
Both solutions satisfy this, so we have the ambiguous case. Let's take B≈45.8° as the acute solution.