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Quizzes > High School Quizzes > Mathematics

Law of Cosines Practice Quiz

Boost your geometry skills with practice problems

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting a high school trigonometry quiz called Cosine Law Challenge.

What is the correct formula for the Law of Cosines for a triangle with sides a, b, and c and an angle C opposite side c?
c² = a² - b² - 2ab cos C
c² = b² + c² - 2ac cos B
c² = a² + b² + 2ab cos C
c² = a² + b² - 2ab cos C
The Law of Cosines formula is expressed as c² = a² + b² - 2ab cos C, which generalizes the Pythagorean theorem for any triangle. This formula directly relates the lengths of the sides with the cosine of the included angle.
When is the Law of Cosines most appropriately used in solving a triangle?
When given two sides and the included angle to find the third side
When using properties of similar triangles
Only when the triangle is right-angled
When only one side and one angle are known
The Law of Cosines is especially useful for solving triangles when two sides and the included angle (SAS) are known, but it also works for solving triangles with three known sides (SSS). It is not limited to right-angled triangles.
In the formula c² = a² + b² - 2ab cos C, which side is represented by c?
The side adjacent to angle C
The side opposite the angle C
The longest side in any triangle
Either of the two sides forming angle C
In the Law of Cosines, c is defined as the side opposite the angle C. This distinction is important for correctly applying the formula.
What is the value of cos 60°?
1
0.5
-0.5
0
The cosine of 60° is 0.5, which is a fundamental trigonometric value. This value is often used when applying the Law of Cosines.
For a triangle with sides of length 5 and 6 and an included angle of 60°, which expression correctly represents the square of the third side?
c² = 5² + 6² - 2*5*6*cos(60°)
c² = 5² + 6² + 2*5*6*cos(60°)
c² = 5² + 6² - cos(60°)
c² = 5² - 6² - 2*5*6*cos(60°)
Substituting the given values into the Law of Cosines formula yields c² = 5² + 6² - 2*5*6*cos(60°). This expression correctly represents the square of the third side.
If a triangle has sides a = 8, b = 5 and an included angle C = 60°, what is the length of side c?
8
6
9
7
Substitute the values into the Law of Cosines: c² = 8² + 5² - 2*8*5*cos(60°). Since cos(60°) = 0.5, we compute c² = 64 + 25 - 80*0.5 = 49, so c = 7.
For a triangle with two sides measuring 9 and 12 and an included angle of 120°, which expression correctly represents the square of the third side?
c² = 9² + 12² - 2*9*12*cos(120°)
c² = 9² - 12² - 2*9*12*cos(120°)
c² = 9² + 12² + 2*9*12*cos(120°)
c² = 9² + 12² - cos(120°)
The Law of Cosines states that c² = a² + b² - 2ab*cos(C). Substituting a = 9, b = 12, and C = 120° provides the correct expression.
For a triangle with sides a = 7, b = 10, and an included angle C = 45°, what is the correct expression for side c?
c = 7² + 10² - 2*7*10*cos(45°)
c = √(7² + 10² - 2*7*10*cos(45°))
c = √(7² + 10² + 2*7*10*cos(45°))
c = 7 + 10 - cos(45°)
The Law of Cosines gives c² = 7² + 10² - 2*7*10*cos(45°). Taking the square root of both sides results in c = √(7² + 10² - 2*7*10*cos(45°)).
What is the correct process to determine an unknown angle using the Law of Cosines?
Isolate the cosine term and apply the inverse cosine function
Take the square root of both sides and then use the sine function
Rearrange the equation to isolate the side lengths
Set the Law of Cosines equal to zero and solve for the angle
To find an unknown angle, the Law of Cosines is rearranged to isolate the cosine term, after which the inverse cosine function is used to determine the angle. This method directly relates the sides to the angle measure.
How does the Law of Cosines simplify for a right-angled triangle?
It simplifies to the Pythagorean theorem
It does not apply to right-angled triangles
It becomes a quadratic equation
It transforms into the Law of Sines
In a right-angled triangle, the cosine of 90° is 0. Therefore, the Law of Cosines reduces to a² + b² = c², which is the familiar Pythagorean theorem.
What is the correct expression for cos C using the Law of Cosines in a triangle with sides a, b, and c, where angle C is opposite side c?
cos C = (a² - b² + c²) / (2ab)
cos C = (a² + b² - c²) / (2ab)
cos C = (b² + c² - a²) / (2bc)
cos C = (a² + c² - b²) / (2ac)
Rearranging the Law of Cosines for angle C yields cos C = (a² + b² - c²) / (2ab), which correctly expresses the cosine of the angle in terms of the three sides.
In triangle DEF with sides d = 13, e = 14 and an included angle of 60° opposite side f, which of the following represents the correct Law of Cosines equation?
f² = 13² + 14² - cos(60°)
f² = 13² + 14² + 2*13*14*cos(60°)
f² = 13² - 14² - 2*13*14*cos(60°)
f² = 13² + 14² - 2*13*14*cos(60°)
Applying the Law of Cosines in triangle DEF, the correct setup is f² = d² + e² - 2de*cos(60°). Substituting d = 13 and e = 14 gives the right expression.
What effect does an obtuse angle have on the Law of Cosines calculation?
It invalidates the use of the Law of Cosines
It has no impact on the calculation
The negative cosine value increases the calculated side length
It decreases the calculated side length
An obtuse angle yields a negative cosine value. When substituted into the Law of Cosines, the term -2ab*cos(C) effectively adds a positive value, increasing the computed side length.
Using the Law of Cosines for a triangle with sides 12, 15, and 9, what is the cosine of the angle opposite the side of length 9?
cos C = (15² + 9² - 12²) / (2*15*9)
cos C = (12² + 9² - 15²) / (2*12*9)
cos C = (12² + 15² - 9²) / (2*12*15)
cos C = (12² + 15² + 9²) / (2*12*15)
By applying the Law of Cosines, we find cos C = (12² + 15² - 9²) / (2*12*15), which is the correct formula for the cosine of the angle opposite the side of length 9.
For a triangle with sides p = 10, q = 15 and an angle of 100° opposite side r, which expression correctly represents r²?
r² = 10² + 15² + 2*10*15*cos(100°)
r² = 10² - 15² + 2*10*15*cos(100°)
r² = 10² - 15² - 2*10*15*cos(100°)
r² = 10² + 15² - 2*10*15*cos(100°)
Substituting the given values into the Law of Cosines, we obtain r² = 10² + 15² - 2*10*15*cos(100°), which is the correct expression for the square of side r.
What is the correct expression for angle A in a triangle with sides a, b, and c using the Law of Cosines?
A = arccos((b² + c² - a²) / (2bc))
A = arcsin((b² + c² - a²) / (2bc))
A = arccos((a² + c² - b²) / (2ac))
A = arccos((a² + b² - c²) / (2ab))
Rearranging the Law of Cosines for angle A gives A = arccos((b² + c² - a²) / (2bc)). This expression correctly relates angle A to the lengths of the sides.
In a triangle with sides 11, 13, and k, if the angle opposite side k is 60°, what is the value of k?
7√2
8√3
13√3
7√3
Using the Law of Cosines: k² = 11² + 13² - 2*11*13*cos(60°). Since cos(60°) is 0.5, the equation simplifies to k² = 121 + 169 - 143 = 147, so k = √147 = 7√3.
For triangle ABC with a = 9, b = 12, and angle C = 135°, calculate side c using the Law of Cosines and then determine angle A using the Law of Sines. Which approximation is closest to angle A?
Approximately 45°
Approximately 19°
Approximately 30°
Approximately 25°
First, the Law of Cosines is used to determine side c, and then the Law of Sines finds angle A by relating a/sin A = c/sin C. The calculations approximate angle A to be around 19°, demonstrating the combined use of both laws.
Using the Law of Cosines for triangle ABC with sides a = 8, b = 15, and c = 17, what is the value of cos A?
15/17
2/3
8/17
17/15
By applying the Law of Cosines for angle A, cos A is given by (b² + c² - a²) / (2bc). Substituting the values yields cos A = (225 + 289 - 64) / (2*15*17) = 450/510, which simplifies to 15/17.
For a triangle with sides 6, 8, and x, where the angle between the sides of lengths 6 and 8 is 120°, determine the length of x and identify the largest angle in the triangle.
x = √100 and the largest angle is 90°
x = √100 and the largest angle is 120°
x = √148 and the largest angle is 120°
x = √148 and the largest angle is 90°
Applying the Law of Cosines with a 120° included angle gives x² = 6² + 8² - 2*6*8*cos(120°) = 148, so x = √148. Since the given angle of 120° is larger than any other angle computed for this triangle, it remains the largest.
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Study Outcomes

  1. Apply the cosine rule to solve triangle problems.
  2. Analyze triangle configurations to determine missing side lengths and angles.
  3. Evaluate different cases where the cosine rule is applicable.
  4. Simplify trigonometric expressions within cosine rule contexts.
  5. Interpret problem statements to identify appropriate strategies for using the cosine rule.

Law of Cosines Worksheet Cheat Sheet

  1. Master the Law of Cosines formula - Get cozy with c² = a² + b² − 2ab cos(C), your secret weapon for non‑right triangles! This equation is like the Pythagorean Theorem's cooler cousin - it handles any triangle when you know two sides and the angle between them. Dive into the Law of Cosines
  2. Spot the right moment to use it - The Law of Cosines shines when you've got two sides and their included angle (SAS) or all three sides (SSS). Memorize those scenarios and you'll instantly know when to reach for this formula instead of the Law of Sines or Pythagoras. Practice with SAS & SSS problems
  3. Plug in values to find missing sides - Treat the formula like a puzzle: plug known side lengths and angle, then solve for the unknown side step by step. Each successful calculation builds your triangle‑solving confidence - kind of like leveling up in a game! Hands‑on Law of Cosines practice
  4. Rearrange it to find angles - Flip the formula into cos(C) = (a² + b² − c²)/(2ab) and hit your calculator's inverse cosine (arccos) button. Before you know it, you'll be uncovering hidden angles like a trigonometry detective. Angle-finding challenges
  5. Connect it to the Pythagorean Theorem - Notice how when C = 90°, the −2ab cos(C) term goes to zero, and bam - you have a² + b² = c². It's the ultimate "aha!" moment that shows these two math legends are really part of the same family. See the Pythagorean link
  6. Apply it in real‑world scenarios - From plotting a hiking trail to surveying a property boundary, the Law of Cosines solves practical navigation and engineering puzzles. Seeing math at work in real life is both inspiring and incredibly satisfying! Real-world problem sets
  7. Combine it with the Law of Sines - When triangles throw mixed info your way, switch between the Law of Cosines and the Law of Sines to crack the code. This dynamic duo ensures you can conquer any triangular challenge. Law of Sines & Cosines worksheets
  8. Create a memory hack - Visualize the Law of Cosines as Pythagoras plus an "angle adjustment" term. Anchoring new formulas to familiar ones makes them stick in your brain like your favorite song. Mnemonic strategies
  9. Drill mixed practice problems - The more triangles you solve - finding sides one minute and angles the next - the more intuitive the Law of Cosines becomes. Embrace variety to transform confusion into confidence. Mixed practice drills
  10. Explore the derivation - Peek under the hood to see how dropping a perpendicular and using the Pythagorean Theorem leads straight to the Law of Cosines. Understanding the proof deepens your grasp and makes the formula truly yours. Derivation explained
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