Practice Law of Sines and Cosines: Trigonometry Unit 7.
This unit covers law of sines, ambiguous case and law of cosines — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers law of sines, ambiguous case and law of cosines — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Law of Sines
The Law of Sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). Use it when you know AAS, ASA, or SSA (with caution on SSA). It is used to find missing sides or angles in non-right triangles.
Key Points
- Formula: a/sin(A) = b/sin(B) = c/sin(C) — can also be written as sin(A)/a = sin(B)/b = sin(C)/c
- Apply when given: two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA)
- Always check that angles sum to 180° when solving for a missing angle
- Use the inverse sine (sin⁻¹) to find a missing angle, and be alert to the ambiguous case with SSA
In triangle ABC, angle A = 35°, angle B = 75°, and side a = 12. Find side b.
First find angle C: 180° − 35° − 75° = 70°. Set up the ratio: 12/sin(35°) = b/sin(75°). Solve for b: b = 12 × sin(75°)/sin(35°) ≈ 12 × 0.9659/0.5736 ≈ 20.2.
2 Ambiguous Case (SSA)
The ambiguous case occurs when given two sides and a non-included angle (SSA), which can produce zero, one, or two valid triangles. Students must determine which scenario applies by comparing the given side opposite the given angle to the height of the triangle (h = b·sin(A)). This is the most frequently tested edge case for Law of Sines.
Key Points
- Given sides a, b and angle A (where a is opposite A): compute h = b·sin(A)
- If a < h → no triangle; if a = h → exactly one right triangle; if h < a < b → two triangles; if a ≥ b → one triangle
- When two triangles exist, the second angle B₂ = 180° − B₁, and both solutions must be checked to ensure all angles stay positive and sum to 180°
- Exams often ask 'how many triangles are possible?' — always show your comparison work
Given a = 10, b = 14, A = 38°. Determine how many triangles exist and find all possible values of angle B.
Compute h = 14·sin(38°) ≈ 14 × 0.6157 ≈ 8.62. Since h ≈ 8.62 < a = 10 < b = 14, two triangles exist. Using Law of Sines: sin(B) = 14·sin(38°)/10 ≈ 0.8619, so B₁ ≈ 59.5° and B₂ ≈ 180° − 59.5° = 120.5°. Both give valid triangles since A + B < 180° in each case.
3 Law of Cosines
The Law of Cosines relates all three sides of a triangle to one of its angles: c² = a² + b² − 2ab·cos(C). Use it when given SAS (two sides and the included angle) or SSS (all three sides). It generalizes the Pythagorean theorem and is the required tool when Law of Sines cannot be applied directly.
Key Points
- Three equivalent forms: a² = b² + c² − 2bc·cos(A); b² = a² + c² − 2ac·cos(B); c² = a² + b² − 2ab·cos(C)
- To find an angle from SSS, rearrange: cos(A) = (b² + c² − a²) / (2bc)
- Use when given SAS or SSS — Law of Sines cannot start these cases without first finding another element
- If the result of cos(A) is negative, the angle is obtuse (between 90° and 180°)
In triangle ABC, a = 8, b = 11, C = 60°. Find side c.
Apply the formula: c² = 8² + 11² − 2(8)(11)·cos(60°) = 64 + 121 − 176·(0.5) = 185 − 88 = 97. Therefore c = √97 ≈ 9.85. After finding c, you could then use Law of Sines to find the remaining angles if needed.
Questions, answered.
What is Law of Sines and Cosines?
Law of Sines and Cosines is Unit 7 of Trigonometry, covering law of sines, ambiguous case and law of cosines.
How to study for Trigonometry Unit 7?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 27+ review questions across 5 different game modes.