Mathematics
Grade 11
15 min
Solve a triangle
Solve a triangle
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1
Introduction & Learning Objectives
Learning Objectives
Identify the given information in a triangle problem as SSS, SAS, ASA, AAS, or the ambiguous SSA case.
Apply the Law of Sines to find unknown side lengths and angle measures.
Apply the Law of Cosines to find unknown side lengths and angle measures.
Use the triangle angle sum theorem in conjunction with trigonometric laws to solve for all unknown parts of a triangle.
Analyze the ambiguous case (SSA) to determine if zero, one, or two triangles exist and solve accordingly.
Calculate the area of an oblique triangle using the formula Area = (1/2)ab sin(C).
Model and solve real-world problems involving triangles.
How do astronomers calculate the vast distance to a star without ever leaving Earth? 🔠They use trigonometry to solve a massive triangle!
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Key Concepts & Vocabulary
TermDefinitionExample
Solve a TriangleThe process of finding the measurements of all three sides and all three angles of a triangle when some of these measurements are given.Given side a = 5, side b = 7, and angle C = 45°, solving the triangle means finding the length of side c and the measures of angles A and B.
Oblique TriangleA triangle that does not have a right angle (90°). The standard SOH CAH TOA rules do not directly apply.A triangle with angles 40°, 60°, and 80° is an oblique triangle.
Included Angle / SideAn 'included angle' is the angle formed between two specified sides. An 'included side' is the side between two specified angles.In triangle ABC, angle C is the included angle between sides a and b. Side c is the included side between angles A and B.
Law of...
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Core Formulas
Law of Sines
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
Use when you know a side and its opposite angle (a 'pair'), plus one other piece of information. Ideal for cases ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and SSA (Side-Side-Angle, the ambiguous case).
Law of Cosines
c^2 = a^2 + b^2 - 2ab \cos(C) \\ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
Use when you do not know a side and its opposite angle. Ideal for cases SAS (Side-Angle-Side) where you need to find the third side, and SSS (Side-Side-Side) where you need to find an angle.
Triangle Angle Sum Theorem
A + B + C = 180°
A fundamental rule used in every problem. Once you know any two angles in a triangle, you can always find the third by subtracting their sum from 180°.
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Sign Up Free to ContinueSample Practice Questions
Easy
A triangle is defined by side lengths a = 8, b = 10, and c = 12. Which case for solving a triangle does this represent?
A.SAS (Side-Angle-Side)
B.ASA (Angle-Side-Angle)
C.SSS (Side-Side-Side)
D.SSA (Side-Side-Angle)
Easy
To solve a triangle where you are given two sides and the included angle (SAS), which law should you apply first?
A.Law of Sines
B.Law of Cosines
C.Pythagorean Theorem
D.Triangle Angle Sum Theorem
Easy
Which of the following correctly represents the Law of Sines for a triangle ABC with sides a, b, and c?
A.a/sin(B) = b/sin(A) = c/sin(C)
B.c^2 = a^2 + b^2 - 2ab cos(C)
C.a/sin(A) = b/sin(B) = c/sin(C)
D.sin(A)/a = sin(B)/b = c/sin(C)
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