Law of Sines/Cosines

Grade 11 · trigonometry · 96 practice problems · read aloud

🔊 Listen to this explanation

Law of Sines & Cosines: Solving Non-Right Triangles

🔍 What Is It & Why Use It?

These laws are your toolkit for solving triangles that do not have a right angle (90°). While SOH-CAH-TOA only works for right triangles, the Law of Sines and Cosines work for any triangle. Use them to find missing side lengths and angles when you know:

  • Law of Sines: Two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA - the ambiguous case).
  • Law of Cosines: Two sides and the included angle (SAS) or all three sides (SSS).

📝 Step-by-Step Guides

Law of Sines

The formula is: a/sin(A) = b/sin(B) = c/sin(C)

  1. Identify the known angle-side pair.
  2. Set up a proportion with the unknown.
  3. Cross-multiply and solve.
Law of Cosines

The formulas are: a² = b² + c² - 2bc·cos(A) (and its variations for b² and c²)

  1. Identify the known parts (SAS or SSS).
  2. Substitute the known values into the correct formula.
  3. Solve for the unknown, using the inverse cosine if finding an angle.

🧮 Worked Examples

Example 1: Law of Sines (AAS)

In ΔABC, angle A = 40°, angle B = 60°, and side a = 10 cm. Find side b.

  1. Set up: 10 / sin(40°) = b / sin(60°)
  2. Cross-multiply: b * sin(40°) = 10 * sin(60°)
  3. Solve: b = (10 * sin(60°)) / sin(40°) ≈ 13.5 cm

Example 2: Law of Cosines (SAS)

In ΔXYZ, side x = 7 m, side y = 5 m, and included angle Z = 85°. Find side z.

  1. Use: z² = x² + y² - 2xy·cos(Z)
  2. Substitute: z² = 7² + 5² - 2(7)(5)·cos(85°)
  3. Calculate: z² ≈ 49 + 25 - 70*(0.0872) ≈ 67.9
  4. Final: z ≈ √67.9 ≈ 8.24 m

⚠️ Common Mistakes

  • Using the wrong law: Remember, Law of Sines needs a known angle-side pair. Law of Cosines is for SAS or SSS.
  • Ambiguous Case (SSA): When given two sides and a non-included angle, there can be 0, 1, or 2 possible triangles. Always check if a second solution exists (the supplementary angle).
  • Calculator mode: Always ensure your calculator is in DEGREE mode, not radians!

💡 Tips & Tricks

  • Memory Aid: For Law of Cosines, it looks like the Pythagorean Theorem with a "correction term" for the non-right angle: "-2bc·cos(A)".
  • Strategy: Use Law of Cosines first when you have SAS or SSS. Use Law of Sines when you have a clear angle-side pair.
  • Angle Check: The largest side is always opposite the largest angle. Use this to check if your answer is reasonable.

🎯 Practice Suggestions

  • Start by categorizing problems as AAS, ASA, SSA, SAS, or SSS before you begin solving.
  • Draw a diagram for every single problem! Label all known parts.
  • Mix up your practice. Do a Law of Sines problem, then a Law of Cosines problem, to force your brain to choose the correct tool.
  • Challenge yourself with word problems that require you to draw the triangle from a description.

Practice problems

6 of the 96, worked through step by step — try them before opening the answer.

1 sin(30°) × 8 = ?

Hint: Recall the exact value of the trigonometric function for common angles and apply it to the coefficient.

Show the answer

Answer: 4

  1. Recall the value of sin(30°). The sine of 30 degrees is a standard trigonometric value: sin(30°) = 1/2.
  2. Write the given multiplication. The problem is: sin(30°) × 8. Substitute the value from
  3. (1/2) × 8.
  4. Perform the multiplication. (1/2) × 8 = 8/2.
  5. Simplify the fraction. 8 divided by 2 equals 4. Final Answer: 4

2 sin(45°) × √2 = ?

Hint: Consider the exact trigonometric value for this common angle and how it relates to the radical expression.

Show the answer

Answer: 1

  1. Recall the value of sin(45°). sin(45°) = √2 / 2.
  2. Write the original expression with this value. sin(45°) × √2 = (√2 / 2) × √2.
  3. Multiply the terms. (√2 / 2) × √2 = (√2 × √2) / 2.
  4. Simplify √2 × √2. √2 × √2 = 2.
  5. Substitute back into the expression. (√2 × √2) / 2 = 2 / 2.
  6. Simplify the fraction. 2 / 2 = 1. Final Answer: 1

3 sin(45°) / 7 = sin(B) / 10

Hint: This equation involves trigonometric ratios and requires isolating the unknown angle. Consider which trigonometric law relates sides and angles in a triangle when you have a known angle-side pair and another side.

Show the answer

Answer: B ≈ 73.4°

The Law of Sines states that in any triangle, the ratio of the sine of an angle to the length of its opposite side is constant. To solve for an unknown angle, you would cross-multiply and then use the inverse sine function. Remember that the sine function has a range where it's one-to-one, so your calculator will give the principal value, which is appropriate for a triangle angle.

4 sin(30°) × 8 ÷ sin(45°) = ?

Hint: Apply the Law of Sines relationship between angles and their opposite sides. Remember to use exact trigonometric values for standard angles.

Show the answer

Answer: 4√2

  1. Recall the exact trigonometric values. We know: sin(30°) = 1/2 sin(45°) = √2 / 2
  2. Substitute these values into the expression. The problem is: sin(30°) × 8 ÷ sin(45°) Substituting gives: (1/2) × 8 ÷ (√2 / 2)
  3. Simplify the multiplication and division. First, multiply (1/2) by 8. (1/2) × 8 = 8/2 = 4 So the expression becomes: 4 ÷ (√2 / 2)
  4. Divide by a fraction. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of (√2 / 2) is (2 / √2). So, 4 ÷ (√2 / 2) = 4 × (2 / √2)
  5. Multiply the numbers. 4 × (2 / √2) = (4 × 2) / √2 = 8 / √2
  6. Rationalize the denominator. It is standard to remove the square root from the denominator. Multiply both the numerator and denominator by √2: (8 / √2) × (√2 / √2) = (8 × √2) / (√2 × √2) = (8√2) / 2
  7. Simplify the fraction. (8√2) / 2 = 4√2 Final Answer: 4√2

5 sin(60°) × 12 ÷ sin(45°) = ?

Hint: Recall the exact values of trigonometric functions for standard angles and apply the Law of Sines relationship.

Show the answer

Answer: 6√6

  1. Recall exact values: sin(60°) = √3/2, sin(45°) = √2/2
  2. Substitute values: (√3/2) × 12 ÷ (√2/2)
  3. Division by fraction is multiplication by reciprocal: (√3/2) × 12 × (2/√2)
  4. Simplify: (√3 × 12 × 2) / (2 × √2) = (√3 × 12) / √2
  5. Rationalize: (12√3) / √2 = (12√3 × √2) / (√2 × √2) = (12√6) / 2
  6. Simplify: 6√6

The answer is 6√6.

6 sin(30°) × 12 ÷ sin(45°) = ?

Hint: Use the Law of Sines relationship to find the unknown side length when given two angles and one side.

Show the answer

Answer: 6√2

  1. Write the expression: sin(30°) × 12 ÷ sin(45°)
  2. Substitute known values: sin(30°) = 1/2, sin(45°) = √2/2
  3. Substitute: (1/2) × 12 ÷ (√2/2)
  4. Simplify: (1/2) × 12 = 6
  5. Now we have 6 ÷ (√2/2) = 6 × (2/√2)
  6. Simplify: 6 × (2/√2) = 12/√2
  7. Rationalize the denominator: (12/√2) × (√2/√2) = 12√2/2
  8. Simplify: 12√2/2 = 6√2

The answer is 6√2.

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