Complete trigonometry guide • Step-by-step solutions
\( c^2 = a^2 + b^2 - 2ab\cos(C) \)
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem to any triangle, not just right triangles. This formula allows you to find the third side of a triangle when you know two sides and the included angle, or to find an angle when you know all three sides.
Key applications include:
When angle C = 90°, cos(C) = 0, and the Law of Cosines reduces to the familiar Pythagorean theorem: c² = a² + b². This makes it a versatile tool for solving any triangle, whether acute, right, or obtuse.
The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. Unlike the Pythagorean theorem, which only applies to right triangles, the Law of Cosines applies to any triangle. It's particularly useful for solving oblique triangles (triangles that are not right triangles) when you know certain combinations of sides and angles.
The Law of Cosines states that for any triangle with sides a, b, c and opposite angles A, B, C respectively:
Or equivalently:
Key characteristics of triangles solved with the Law of Cosines:
c² = a² + b² - 2ab cos(C), where a, b, c are sides and C is the angle opposite side c.
\(c^2 = a^2 + b^2 - 2ab\cos(C)\)
Where a, b, c = sides of triangle, C = angle opposite side c.
Used for oblique triangles, navigation, surveying, and engineering problems.
In triangle ABC, side a = 8, side b = 6, and angle C = 60°. What is the length of side c?
Using the Law of Cosines: c² = a² + b² - 2ab cos(C)
Given: a = 8, b = 6, C = 60°
Step 1: Calculate cos(60°) = 0.5
Step 2: Substitute values: c² = 8² + 6² - 2(8)(6)cos(60°)
Step 3: c² = 64 + 36 - 96(0.5) = 100 - 48 = 52
Step 4: c = √52 ≈ 7.21
The answer is B) 7.21.
The Law of Cosines is particularly useful when we have two sides and the included angle (SAS case). This is exactly what we have here: sides a and b, and the angle C between them. The formula c² = a² + b² - 2ab cos(C) allows us to find the third side directly. Notice how the Law of Cosines generalizes the Pythagorean theorem: when angle C is 90°, cos(90°) = 0, and the formula reduces to c² = a² + b².
Law of Cosines: Relates sides and angles in any triangle
SAS Case: Two sides and included angle known
Included Angle: The angle between two known sides
• c² = a² + b² - 2ab cos(C) (when C is between sides a and b)
• cos(60°) = 0.5
• Always take the positive square root for side length
• Remember: the angle in the formula is between the two known sides
• Double-check your cosine value (especially for common angles)
• The result should make sense compared to the known sides
• Using the wrong angle in the formula (not the included angle)
• Forgetting to subtract 2ab cos(C) instead of adding
• Using the Law of Sines instead of Cosines when you have SAS
A triangle has sides of length 7, 10, and 12. Find the measure of the angle opposite the side of length 12. Round to the nearest tenth of a degree.
We'll use the rearranged Law of Cosines to find an angle: cos(C) = (a² + b² - c²) / (2ab)
Given: a = 7, b = 10, c = 12 (where C is opposite side c)
Step 1: Substitute values: cos(C) = (7² + 10² - 12²) / (2×7×10)
Step 2: cos(C) = (49 + 100 - 144) / 140 = 5 / 140 = 1/28
Step 3: C = arccos(1/28) ≈ 87.9°
Therefore, the angle opposite the side of length 12 is approximately 87.9°.
When we know all three sides of a triangle (SSS case), we can use the Law of Cosines to find any angle. We rearrange the formula to solve for the cosine of the angle: cos(C) = (a² + b² - c²) / (2ab). This version is particularly useful because it allows us to find an angle when all sides are known. The angle found is always the one opposite to the side whose length appears in the numerator as the subtracted term.
SSS Case: All three sides of a triangle are known
Opposite Side: The side across from a given angle
Arccosine: The inverse cosine function (cos⁻¹)
• cos(C) = (a² + b² - c²) / (2ab) when C is opposite side c
• Always check that the angle is between 0° and 180°
• The largest angle is opposite the longest side
• Remember: the angle you're finding is opposite the side in the denominator
• Double-check your arithmetic when calculating (a² + b² - c²)
• Use a calculator for the arccosine function
• Mixing up the order of sides in the formula
• Forgetting to divide by 2ab in the denominator
• Getting a domain error because of incorrect arithmetic
A ship travels 15 km east, then turns and travels 20 km in a direction that forms a 120° angle with its original course. How far is the ship from its starting point? Round to the nearest tenth of a kilometer.
Step 1: Model as a triangle with sides a = 15 km, b = 20 km, and included angle C = 120°
Step 2: Use Law of Cosines: c² = a² + b² - 2ab cos(C)
Step 3: Calculate cos(120°) = -0.5
Step 4: Substitute: c² = 15² + 20² - 2(15)(20)cos(120°)
Step 5: c² = 225 + 400 - 600(-0.5) = 625 + 300 = 925
Step 6: c = √925 ≈ 30.4 km
Therefore, the ship is approximately 30.4 km from its starting point.
This is a classic navigation problem that demonstrates the practical application of the Law of Cosines. The ship's journey forms a triangle where we know two sides (the distances traveled) and the angle between them. The key insight is recognizing that the angle of 120° is the interior angle of the triangle formed by the ship's path. When dealing with obtuse angles (like 120°), remember that cos(120°) is negative, which adds to the sum of squares rather than subtracting.
Navigation Problem: Real-world scenario involving distances and directions
Interior Angle: The angle inside the triangle formed by the paths
Obtuse Angle: An angle greater than 90° but less than 180°
• cos(120°) = -0.5 (negative for obtuse angles)
• When cos(C) is negative, 2ab cos(C) becomes addition
• The Law of Cosines works for any angle between 0° and 180°
• Draw a diagram to visualize the triangle formed by the paths
• Pay attention to whether the angle is acute or obtuse
• Remember that cos(angle) is negative for obtuse angles
• Forgetting that cos(120°) is negative
• Adding instead of subtracting 2ab cos(C) when cos(C) is negative
• Misidentifying the angle between the two known sides
An engineer needs to determine the length of a support beam for a triangular roof truss. Two sides of the triangle measure 12 feet and 16 feet, and the angle between them is 75°. Calculate the length of the third side (the support beam). Also determine if this angle makes the triangle acute, right, or obtuse.
Using the Law of Cosines with a = 12, b = 16, and C = 75°:
Step 1: Calculate cos(75°) ≈ 0.2588
Step 2: Apply formula: c² = 12² + 16² - 2(12)(16)cos(75°)
Step 3: c² = 144 + 256 - 384(0.2588) = 400 - 99.38 ≈ 300.62
Step 4: c = √300.62 ≈ 17.34 feet
For the classification: Since C = 75° < 90°, this is an acute angle. To confirm the entire triangle is acute, we'd need to check all angles, but knowing one angle is acute doesn't guarantee the triangle is acute.
The support beam should be approximately 17.34 feet long.
This engineering application shows how the Law of Cosines is used in construction and structural design. When designing triangular supports, engineers often know two sides and the angle between them, making the Law of Cosines the perfect tool. The classification of triangles as acute, right, or obtuse is important for structural integrity. A triangle is acute if all angles are less than 90°, right if one angle is exactly 90°, and obtuse if one angle is greater than 90°.
Roof Truss: Framework of beams supporting a roof structure
Support Beam: Structural element providing support in a frameworkAcute Triangle: All angles are less than 90°
• cos(75°) ≈ 0.2588 (positive for acute angles)
• Triangle classifications depend on all angles
• The Law of Cosines is essential in structural engineering
• For common angles like 75°, use calculator for precise cosine value
• In engineering, always round to appropriate precision for construction
• Consider safety factors in real-world applications
• Using the wrong angle in the formula (not the included angle)
• Approximating cosine values too roughly for precision work
• Confusing triangle classification with individual angle classification
In triangle ABC, side lengths are a = 7, b = 10, and c = 12. Which of the following statements about angle C (opposite to side c) is correct?
Using the Law of Cosines rearranged to find the angle: cos(C) = (a² + b² - c²) / (2ab)
Given: a = 7, b = 10, c = 12
Step 1: cos(C) = (7² + 10² - 12²) / (2×7×10) = (49 + 100 - 144) / 140 = 5/140 = 1/28
Step 2: cos(C) = 1/28 ≈ 0.0357
Step 3: Since cos(C) > 0, angle C is acute (less than 90°)
Step 4: C = arccos(1/28) ≈ 87.9°
The answer is A) C is acute, measuring less than 90°.
We can determine if an angle is acute, right, or obtuse by examining the sign of its cosine value. If cos(C) > 0, then C is acute (between 0° and 90°). If cos(C) = 0, then C is right (exactly 90°). If cos(C) < 0, then C is obtuse (between 90° and 180°). In this case, since we got cos(C) = 1/28 ≈ 0.0357 > 0, the angle is acute. This is consistent with the fact that in any triangle, the largest angle is opposite the longest side, but as long as the triangle inequality holds, all angles can be acute.
Acute Angle: An angle less than 90°
Right Angle: An angle exactly 90°
Obtuse Angle: An angle greater than 90° but less than 180°
• cos(C) > 0 → C is acute
• cos(C) = 0 → C is right
• cos(C) < 0 → C is obtuse
• Just by comparing cos(C) to 0, you can classify the angle
• The longest side is always opposite the largest angle
• Use the rearranged Law of Cosines to find angles efficiently
• Confusing the sign of cosine with angle classification
• Forgetting that cos(90°) = 0
• Making calculation errors in the numerator (a² + b² - c²)
Q: When should I use the Law of Cosines instead of the Law of Sines?
A: The Law of Cosines is preferred in two specific scenarios:
1. SAS (Side-Angle-Side): When you know two sides and the included angle (the angle between those sides). For example, if you know sides a and b, and the angle C between them, use: c² = a² + b² - 2ab cos(C).
2. SSS (Side-Side-Side): When you know all three sides of the triangle and want to find an angle. In this case, rearrange the formula: cos(C) = (a² + b² - c²) / (2ab).
The Law of Sines works best for ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side) situations. The Law of Cosines is also preferred when you encounter the ambiguous case (SSA) with the Law of Sines, as it provides a definitive answer.
Q: How does the Law of Cosines relate to the Pythagorean theorem?
A: The Law of Cosines is actually a generalization of the Pythagorean theorem. When we examine the Law of Cosines formula:
c² = a² + b² - 2ab cos(C)
In a right triangle where angle C = 90°, we know that cos(90°) = 0. Substituting this into the formula:
c² = a² + b² - 2ab cos(90°)
c² = a² + b² - 2ab × 0
c² = a² + b²
This is precisely the Pythagorean theorem! So the Law of Cosines extends the Pythagorean theorem to work with any triangle, not just right triangles. The term "-2ab cos(C)" acts as a correction factor that adjusts for the deviation from a right angle.