Unit 8: Laws of Sines and Cosines
Solving triangles that aren't right triangles, the ambiguous case where one setup builds two different triangles, and finding area without a height.
5 topics · 21 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Law of Sines
- How it worksHow does the Law of Sines keep the ratio of a side to the sine of its opposite angle constant across a triangle?
- Worked exampleFind side b in a triangle with angle A = 40 degrees, angle B = 65 degrees, and side a = 12.
- Common mistakeWhy doesn't the Law of Sines only apply to right triangles the way SOH-CAH-TOA does?
- Ties togetherHow does the Law of Sines generalize right-triangle trigonometry to triangles with no right angle at all?
Your turn1 of 2
Write side b as a function of angle B (in radians), given angle A = 55 degrees and side a = 14.The Ambiguous Case (SSA)
- How it worksHow does swinging the side opposite the given angle reveal whether zero, one, or two triangles are possible?
- Worked exampleFind both possible triangles given side a = 10, side b = 14, and angle A = 30 degrees.
- Worked exampleFind whether a triangle exists at all given side a = 6, side b = 15, and angle A = 50 degrees.
- Common mistakeWhy doesn't knowing side-side-angle guarantee a triangle even exists, the way side-angle-side always does?
- Common mistakeWhy doesn't the ambiguous case ever produce two triangles once the given angle reaches 90 degrees or more?
Your turn1 of 2
A student assumes that knowing two sides and a non-included angle (SSA) always determines a triangle just as reliably as knowing two sides and the INCLUDED angle (SAS) does. What's the flaw?The Law of Cosines
- How it worksHow does the Law of Cosines reduce to the Pythagorean theorem when the included angle is exactly 90 degrees?
- Worked exampleFind side c in a triangle with sides a = 8, b = 10, and included angle C = 60 degrees.
- Worked exampleFind the largest angle in a triangle with sides 7, 9, and 13.
- Common mistakeWhy doesn't the Law of Cosines put the angle opposite the side you're solving for on the wrong side of the equation?
Your turn1 of 2
Write side c as a function of angle C (in radians), given sides a = 11 and b = 11.Choosing Between the Law of Sines and the Law of Cosines
- How it worksHow do you tell from a triangle's given sides and angles whether to reach for the Law of Sines or the Law of Cosines?
- Worked exampleFind the missing side or angle for a triangle given all three sides: 5, 7, and 9.
- Worked exampleFind the missing parts of a triangle given two angles and the included side: A = 50 degrees, C = 60 degrees, b = 10.
- Going furtherWhat happens when a triangle gives you enough information to use either law at once?
Your turn1 of 2
A triangle has all three sides p = 11, q = 11, and r = 12. Since all three sides are known, use the Law of Cosines to find the angle opposite side p (in degrees).The Area of a Triangle (SAS and Heron's Formula)
- How it worksHow does the formula (1/2)ab·sin(C) replace the base-times-height formula when no height is given directly?
- Worked exampleFind the area of a triangle with sides 8 and 10 and an included angle of 50 degrees.
- Worked exampleFind the area of a triangle with sides 5, 6, and 7 using Heron's formula.
- Common mistakeWhy doesn't Heron's formula need you to know a single angle of the triangle?
Your turn1 of 2
Write the area of a triangle as a function of the included angle C (in radians), given sides 11 and 11.