The law of cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem, which is a special case of the law of cosines when the triangle is a right triangle.
The law of cosines can be used to solve a variety of problems involving triangles, such as finding the length of a side of a triangle when the other two sides and the angle between them are known, or finding the angle between two sides of a triangle when the other two sides and the angle between them are known.
The law of cosines can be expressed in a variety of ways, but the most common form is:
Law of Cosine Calculator
Powerful triangle solver tool.
- Find side lengths.
- Find angles.
- Solve SAS triangles.
- Solve SSA triangles.
- Solve SSS triangles.
- Step-by-step solutions.
- Free to use.
- Easy to use.
The law of cosine calculator is a valuable tool for students, engineers, and anyone else who works with triangles.
Find side lengths.
The law of cosines can be used to find the length of a side of a triangle when the other two sides and the angle between them are known. This is useful in a variety of situations, such as when you need to find the length of a diagonal of a rectangle or the length of a side of a triangle in a survey.
To find the length of a side of a triangle using the law of cosines, you can use the following formula:
c^2 = a^2 + b^2 - 2ab cos(C)
where: * `c` is the length of the side you are trying to find * `a` and `b` are the lengths of the other two sides of the triangle * `C` is the angle between sides `a` and `b`
To use this formula, simply plug in the values you know and solve for `c`. For example, let's say you have a triangle with sides of length 5 and 7, and the angle between those sides is 60 degrees. To find the length of the third side, you would plug these values into the formula as follows:
c^2 = 5^2 + 7^2 - 2(5)(7) cos(60°)
c^2 = 25 + 49 - 70 cos(60°)
c^2 = 74 - 70(0.5)
c^2 = 74 - 35
c^2 = 39
c = √39
c = 6.245
Therefore, the length of the third side of the triangle is 6.245 units.
The law of cosines can also be used to find the length of a side of a triangle when you know the lengths of the other two sides and the measure of one of the non-included angles. This is useful in situations such as when you need to find the length of a side of a triangle in a survey or when you need to find the length of a diagonal of a parallelogram.
Find angles.
The law of cosines can also be used to find the measure of an angle in a triangle when you know the lengths of the three sides. This is useful in a variety of situations, such as when you need to find the measure of an angle in a survey or when you need to find the measure of an angle in a machine part.
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Angle opposite the known side:
To find the measure of the angle opposite the known side, you can use the following formula:
cos(C) = (a^2 + b^2 - c^2) / 2ab
where: * `C` is the angle opposite the known side * `a` and `b` are the lengths of the other two sides of the triangle * `c` is the length of the known side
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Angle opposite the unknown side:
To find the measure of the angle opposite the unknown side, you can use the following formula:
C = arccos((a^2 + b^2 - c^2) / 2ab)
where: * `C` is the angle opposite the unknown side * `a` and `b` are the lengths of the sides adjacent to the unknown angle * `c` is the length of the side opposite the unknown angle
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Angle between two known sides:
To find the measure of the angle between two known sides, you can use the following formula:
C = arccos((a^2 + b^2 - c^2) / 2ab)
where: * `C` is the angle between the two known sides * `a` and `b` are the lengths of the two known sides * `c` is the length of the third side
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Angle between two unknown sides:
To find the measure of the angle between two unknown sides, you can use the following formula:
C = arccos((a^2 + b^2 - c^2) / 2ab)
where: * `C` is the angle between the two unknown sides * `a`, `b`, and `c` are the lengths of the three sides of the triangle
The law of cosines is a powerful tool for solving a variety of problems involving triangles. It can be used to find the length of a side of a triangle, the measure of an angle in a triangle, and the area of a triangle.
Solve SAS triangles.
A SAS triangle is a triangle in which two sides and the included angle are known. To solve a SAS triangle, you can use the law of cosines to find the length of the third side, and then use the law of sines to find the measures of the other two angles.
To find the length of the third side of a SAS triangle, you can use the following formula:
c^2 = a^2 + b^2 - 2ab cos(C)
where: * `c` is the length of the third side * `a` and `b` are the lengths of the other two sides * `C` is the included angle
Once you know the length of the third side, you can use the law of sines to find the measures of the other two angles. The law of sines states that the ratio of the sine of an angle to the length of the opposite side is the same for all angles in a triangle. In other words:
sin(A) / a = sin(B) / b = sin(C) / c
where: * `A`, `B`, and `C` are the angles of the triangle * `a`, `b`, and `c` are the lengths of the opposite sides
To find the measure of an angle in a SAS triangle, you can use the following formula:
sin(A) = (b sin(C)) / c
where: * `A` is the angle you are trying to find * `b` is the length of the side opposite angle `A` * `C` is the included angle * `c` is the length of the third side
You can also use the law of cosines to find the measures of the other two angles in a SAS triangle. However, this is usually more difficult than using the law of sines.
Here is an example of how to solve a SAS triangle:
Given a SAS triangle with sides of length 5, 7, and an included angle of 60 degrees, find the length of the third side and the measures of the other two angles.
To find the length of the third side, we can use the law of cosines:
c^2 = a^2 + b^2 - 2ab cos(C)
c^2 = 5^2 + 7^2 - 2(5)(7) cos(60°)
c^2 = 25 + 49 - 70 cos(60°)
c^2 = 74 - 70(0.5)
c^2 = 74 - 35
c^2 = 39
c = √39
c = 6.245
Therefore, the length of the third side of the triangle is 6.245 units.
To find the measure of angle A, we can use the law of sines:
sin(A) = (b sin(C)) / c
sin(A) = (7 sin(60°)) / 6.245
sin(A) = (7 * 0.866) / 6.245
sin(A) = 0.999
A = arcsin(0.999)
A = 89.99°
Therefore, the measure of angle A is 89.99 degrees.
To find the measure of angle B, we can use the fact that the sum of the angles in a triangle is 180 degrees:
B = 180° - A - C
B = 180° - 89.99° - 60°
B = 30.01°
Therefore, the measure of angle B is 30.01 degrees.
Solve SSA triangles.
An SSA triangle is a triangle in which two sides and a non-included angle are known. To solve an SSA triangle, you can use the law of cosines to find the length of the third side, and then use the law of sines to find the measures of the other two angles.
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Side opposite the known angle:
To find the length of the side opposite the known angle, you can use the following formula:
c = sqrt(a^2 + b^2 - 2ab cos(C))
where: * `c` is the length of the side opposite the known angle * `a` and `b` are the lengths of the other two sides * `C` is the known angle
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Angle opposite the known side:
To find the measure of the angle opposite the known side, you can use the following formula:
A = arcsin((b sin(C)) / c)
where: * `A` is the angle opposite the known side * `b` is the length of the known side * `C` is the known angle * `c` is the length of the third side
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Angle between the two unknown sides:
To find the measure of the angle between the two unknown sides, you can use the following formula:
B = 180° - A - C
where: * `B` is the angle between the two unknown sides * `A` is the angle opposite the known side * `C` is the known angle
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Using the ambiguous case:
In some cases, there may be two possible solutions for the SSA triangle. This is known as the ambiguous case. To determine if the ambiguous case occurs, you need to calculate the following value:
k = (a^2 + b^2 - c^2) / (2ab)
where: * `a`, `b`, and `c` are the lengths of the sides of the triangle
If `k` is greater than 1, then the ambiguous case occurs. If `k` is less than 1, then there is only one solution for the SSA triangle. If `k` is equal to 1, then the triangle is a right triangle.
Here is an example of how to solve an SSA triangle:
Given an SSA triangle with sides of length 5, 7, and an angle of 60 degrees opposite the side of length 7, find the length of the third side and the measures of the other two angles.
To find the length of the third side, we can use the law of cosines:
c = sqrt(a^2 + b^2 - 2ab cos(C))
c = sqrt(5^2 + 7^2 - 2(5)(7) cos(60°))
c = sqrt(25 + 49 - 70 cos(60°))
c = sqrt(74 - 70(0.5))
c = sqrt(74 - 35)
c = sqrt(39)
c = 6.245
Therefore, the length of the third side of the triangle is 6.245 units.
To find the measure of angle A, we can use the law of sines:
A = arcsin((b sin(C)) / c)
A = arcsin((7 sin(60°)) / 6.245)
A = arcsin((7 * 0.866) / 6.245)
A = arcsin(0.999)
A = 89.99°
Therefore, the measure of angle A is 89.99 degrees.
To find the measure of angle B, we can use the fact that the sum of the angles in a triangle is 180 degrees:
B = 180° - A - C
B = 180° - 89.99° - 60°
B = 30.01°
Therefore, the measure of angle B is 30.01 degrees.
Solve SSS triangles.
An SSS triangle is a triangle in which all three sides are known. To solve an SSS triangle, you can use the law of cosines to find the measures of all three angles.
To find the measure of an angle in an SSS triangle, you can use the following formula:
cos(C) = (a^2 + b^2 - c^2) / 2ab
where: * `C` is the angle you are trying to find * `a`, `b`, and `c` are the lengths of the sides of the triangle
Once you know the measure of one angle, you can use the fact that the sum of the angles in a triangle is 180 degrees to find the measures of the other two angles.
Here is an example of how to solve an SSS triangle:
Given an SSS triangle with sides of length 5, 7, and 8, find the measures of all three angles.
To find the measure of angle C, we can use the law of cosines:
cos(C) = (a^2 + b^2 - c^2) / 2ab
cos(C) = (5^2 + 7^2 - 8^2) / 2(5)(7)
cos(C) = (25 + 49 - 64) / 70
cos(C) = 10 / 70
C = arccos(10 / 70)
C = 82.26°
Therefore, the measure of angle C is 82.26 degrees.
To find the measure of angle A, we can use the fact that the sum of the angles in a triangle is 180 degrees:
A = 180° - B - C
A = 180° - 82.26°
A = 97.74°
Therefore, the measure of angle A is 97.74 degrees.
To find the measure of angle B, we can use the fact that the sum of the angles in a triangle is 180 degrees:
B = 180° - A - C
B = 180° - 97.74° - 82.26°
B = 0.00°
Therefore, the measure of angle B is 0.00 degrees.
Note that in some cases, the law of cosines may not be able to be used to solve an SSS triangle. This can happen if the triangle is very close to being isosceles, or if the triangle is very close to being a right triangle.
Step-by-step solutions.
The law of cosine calculator provides step-by-step solutions to all problems. This means that you can see each step of the calculation, from the initial input to the final answer. This can be helpful for understanding how the law of cosines works and for checking your work.
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Input the values of the known sides and angles.
The first step is to input the values of the known sides and angles into the calculator. Make sure that you enter the values correctly, as any errors will affect the accuracy of the solution.
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Select the type of problem you want to solve.
Once you have entered the known values, you need to select the type of problem you want to solve. The law of cosine calculator can solve a variety of problems, including finding the length of a side, finding the measure of an angle, and finding the area of a triangle.
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Click the "Calculate" button.
Once you have selected the type of problem you want to solve, click the "Calculate" button. The calculator will then perform the necessary calculations and display the solution.
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Review the solution.
Once the calculator has displayed the solution, you should review it carefully to make sure that it is correct. You can also use the step-by-step solution to see how the calculator arrived at the answer.
Here is an example of how to use the law of cosine calculator to solve a problem:
Given a triangle with sides of length 5, 7, and 8, find the measure of angle C.
To solve this problem, you would follow these steps:
- Input the values of the known sides and angles into the calculator.
- Select the "Find the measure of an angle" option.
- Click the "Calculate" button.
- Review the solution.
The calculator would then display the solution, which is that angle C measures 82.26 degrees.
Free to use.
The law of cosine calculator is free to use. This means that you can use it as much as you want, without having to pay any fees. This makes it a great option for students, engineers, and anyone else who needs to solve problems involving triangles.
There are a number of reasons why the law of cosine calculator is free to use. First, it is a relatively simple calculator to develop. Second, the developers of the calculator believe that everyone should have access to the tools they need to solve math problems. Finally, the calculator is supported by advertising, which helps to cover the costs of development and maintenance.
Here are some of the benefits of using a free law of cosine calculator:
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You can save money.
By using a free calculator, you can avoid having to purchase a commercial calculator. This can save you a significant amount of money, especially if you need to use the calculator frequently.
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You can access the calculator from anywhere.
Because the law of cosine calculator is available online, you can access it from anywhere with an internet connection. This means that you can use the calculator at home, at school, or at work.
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You can use the calculator as many times as you want.
There is no limit to the number of times you can use the law of cosine calculator. This means that you can use it to solve as many problems as you need to.
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You can get help from the calculator's developers.
If you have any questions about how to use the calculator, you can contact the developers for help. They will be happy to assist you.
The law of cosine calculator is a valuable tool for anyone who needs to solve problems involving triangles. It is free to use, easy to access, and can be used as many times as needed. If you need to solve a problem involving a triangle, be sure to give the law of cosine calculator a try.
Easy to use.
The law of cosine calculator is easy to use. It has a simple and intuitive interface that makes it easy to input the values of the known sides and angles and select the type of problem you want to solve. The calculator then performs the necessary calculations and displays the solution.
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Clear and concise instructions.
The law of cosine calculator provides clear and concise instructions on how to use the calculator. These instructions are easy to follow, even for those who are not familiar with using calculators.
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Well-organized interface.
The law of cosine calculator has a well-organized interface that makes it easy to find the information you need. The calculator is divided into three sections: the input section, the output section, and the help section. The input section is where you enter the values of the known sides and angles. The output section is where the calculator displays the solution. The help section contains instructions on how to use the calculator and information about the law of cosines.
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No need to install software.
The law of cosine calculator is a web-based calculator. This means that you do not need to install any software on your computer to use it. You can simply access the calculator from your web browser.
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Works on all devices.
The law of cosine calculator is compatible with all devices that have a web browser. This means that you can use the calculator on your computer, your tablet, or your smartphone.
Here are some tips for using the law of cosine calculator:
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Make sure that you enter the values of the known sides and angles correctly.
Any errors in the input will affect the accuracy of the solution.
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Select the type of problem you want to solve carefully.
The law of cosine calculator can solve a variety of problems, so it is important to select the correct type of problem before you click the "Calculate" button.
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Review the solution carefully.
Once the calculator has displayed the solution, you should review it carefully to make sure that it is correct. You can also use the step-by-step solution to see how the calculator arrived at the answer.
FAQ
The law of cosine calculator is a powerful tool that can be used to solve a variety of problems involving triangles. Here are some frequently asked questions about the calculator:
Question 1: What types of problems can the calculator solve?
Answer: The law of cosine calculator can solve a variety of problems involving triangles, including finding the length of a side, finding the measure of an angle, and finding the area of a triangle.
Question 2: How do I use the calculator?
Answer: The law of cosine calculator is easy to use. Simply enter the values of the known sides and angles into the calculator and select the type of problem you want to solve. The calculator will then perform the necessary calculations and display the solution.
Question 3: Is the calculator accurate?
Answer: Yes, the law of cosine calculator is accurate. The calculator uses precise mathematical algorithms to perform the calculations, so you can be confident that the solutions are correct.
Question 4: Is the calculator free to use?
Answer: Yes, the law of cosine calculator is free to use. You can use it as much as you want, without having to pay any fees.
Question 5: Do I need to install any software to use the calculator?
Answer: No, you do not need to install any software to use the law of cosine calculator. The calculator is a web-based calculator, which means that you can access it from any device with a web browser.
Question 6: Can I save my calculations?
Answer: Yes, you can save your calculations using the law of cosine calculator. Simply click the "Save" button to save your calculations to your computer. You can then load your calculations later by clicking the "Load" button.
Question 7: Can I get help using the calculator?
Answer: Yes, you can get help using the law of cosine calculator. The calculator has a built-in help section that provides instructions on how to use the calculator and information about the law of cosines. You can also contact the developers of the calculator for help.
Closing Paragraph for FAQ:
The law of cosine calculator is a valuable tool for anyone who needs to solve problems involving triangles. It is easy to use, accurate, and free to use. If you need to solve a problem involving a triangle, be sure to give the law of cosine calculator a try.
In addition to the FAQ, here are some tips for using the law of cosine calculator:
Tips
Here are some tips for using the law of cosine calculator:
Tip 1: Double-check your input.
Before you click the "Calculate" button, double-check your input to make sure that you have entered the values of the known sides and angles correctly. Any errors in the input will affect the accuracy of the solution.
Tip 2: Select the correct problem type.
The law of cosine calculator can solve a variety of problems involving triangles. Make sure that you select the correct problem type before you click the "Calculate" button. Otherwise, the calculator will not be able to solve the problem correctly.
Tip 3: Review the solution carefully.
Once the calculator has displayed the solution, review it carefully to make sure that it is correct. You can also use the step-by-step solution to see how the calculator arrived at the answer. This can be helpful for understanding how the law of cosines works and for checking your work.
Tip 4: Use the calculator to learn about the law of cosines.
The law of cosine calculator can be a helpful tool for learning about the law of cosines. You can use the calculator to explore different types of problems and to see how the law of cosines is used to solve these problems. This can be a great way to improve your understanding of the law of cosines.
Closing Paragraph for Tips:
By following these tips, you can get the most out of the law of cosine calculator. The calculator is a valuable tool for solving problems involving triangles, and it can also be a helpful tool for learning about the law of cosines.
Now that you know how to use the law of cosine calculator effectively, you can use it to solve a variety of problems involving triangles. Whether you are a student, an engineer, or a mathematician, the law of cosine calculator is a valuable tool that can save you time and effort.
Conclusion
The law of cosine calculator is a valuable tool for anyone who needs to solve problems involving triangles. It is easy to use, accurate, and free to use. The calculator can solve a variety of problems, including finding the length of a side, finding the measure of an angle, and finding the area of a triangle.
Here are some of the main points about the law of cosine calculator:
- It is a free online calculator that can be used to solve a variety of problems involving triangles.
- It is easy to use, with a simple and intuitive interface.
- It is accurate, using precise mathematical algorithms to perform the calculations.
- It can be used to solve a variety of problems, including finding the length of a side, finding the measure of an angle, and finding the area of a triangle.
- It can be used to learn about the law of cosines.
If you need to solve a problem involving a triangle, be sure to give the law of cosine calculator a try. It is a powerful tool that can save you time and effort.
Closing Message:
We hope that you have found this article helpful. If you have any questions about the law of cosine calculator, please feel free to contact us. We are always happy to help.