Mathematics
Grade 10
15 min
Solve a triangle
Solve a triangle
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the type of triangle problem (right-angled vs. non-right-angled) based on given information.
Apply the primary trigonometric ratios (SOH CAH TOA) to find all unknown sides and angles of a right-angled triangle.
Select and apply the Sine Law to solve oblique triangles when given Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS).
Select and apply the Cosine Law to solve oblique triangles when given Side-Angle-Side (SAS) or Side-Side-Side (SSS).
Distinguish between the conditions that require the Sine Law versus the Cosine Law.
Verify the solution to a triangle problem by confirming the sum of angles is 180° and that side-angle relationships are consistent.
Ever wondered how surveyors measure the width of a river without crossing it, or how astronome...
2
Key Concepts & Vocabulary
TermDefinitionExample
Solve a TriangleThe process of finding the measurements of all three angles and all three sides of a triangle. A complete solution consists of 6 values (3 sides, 3 angles).Given a triangle with Angle A = 30°, Angle B = 70°, and side c = 5 cm, 'solving' it means finding Angle C, side a, and side b.
Right-Angled TriangleA triangle containing one 90° angle. The primary trigonometric ratios (SOH CAH TOA) are used exclusively for these triangles.A triangle with angles 30°, 60°, and 90°.
Oblique TriangleA triangle that does not have a 90° angle. It can be acute (all angles < 90°) or obtuse (one angle > 90°). The Sine Law and Cosine Law are used for these.A triangle with angles 50°, 60°, and 70°.
Side-Angle Naming ConventionIn any triangle ABC, the side oppo...
3
Core Formulas
Primary Trigonometric Ratios (SOH CAH TOA)
sin(θ) = Opposite/Hypotenuse
cos(θ) = Adjacent/Hypotenuse
tan(θ) = Opposite/Adjacent
Used ONLY for right-angled triangles to find an unknown side or angle when you know two pieces of information (besides the 90° angle).
The Sine Law
a/sin(A) = b/sin(B) = c/sin(C)
Used for oblique triangles when you know a side and its opposite angle, plus one other piece of information. Ideal for AAS and ASA cases.
The Cosine Law
To find a side: c² = a² + b² - 2ab*cos(C)
To find an angle: cos(C) = (a² + b² - c²)/(2ab)
Used for oblique triangles when you know two sides and the angle between them (SAS) to find the third side, or when you know all three sides (SSS) to find an angle.
5 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Two points, A and B, are on level ground 100 meters apart. A surveyor at point A measures the angle to a landmark C as ∠CAB = 50°. A surveyor at point B measures ∠CBA = 70°. What is the distance from point A to the landmark C?
A.91.0 m
B.108.5 m
C.87.9 m
D.115.2 m
Challenging
In triangle ABC, a=10 and B=40°. Which additional piece of information would require you to start solving with the Cosine Law instead of the Sine Law?
A.Angle A = 60°
B.Angle C = 80°
C.Side b = 15
D.Side c = 12
Challenging
A student is solving for side 'x' in a triangle where x² = 10² + 12² - 2(10)(12)cos(55°). They first calculate 2(10)(12)cos(55°) ≈ 137.65. Then they calculate x² = 100 + 144 - 137.65 = 106.35, so x ≈ 10.31. A second student keeps the full value of cos(55°) in their calculator and gets x ≈ 10.32. What pitfall did the first student fall into?
A.Calculator Mode Error
B.Incorrect Law Selection
C.Rounding Too Early
D.Order of Operations Error
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free