I’m The Law Of Cosines In ΔABC, A = 2.6 Cm, B = 4.3 Cm And C=4.7 Cm. Find The Measure Of ∠A To The Nearest Degree.

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Introduction

The law of cosines is a fundamental concept in trigonometry that allows us to find the length of a side of a triangle when we know the lengths of the other two sides and the measure of the angle between them. In this article, we will explore the law of cosines and use it to solve a problem involving a triangle with known side lengths.

What is the Law of Cosines?

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. The formula is:

c² = a² + b² - 2ab * cos(C)

where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.

How to Use the Law of Cosines

To use the law of cosines, we need to know the lengths of the three sides of the triangle and the measure of one of the angles. We can then plug these values into the formula and solve for the unknown angle.

Example Problem

Let's consider the triangle ΔABC, where a = 2.6 cm, b = 4.3 cm, and c = 4.7 cm. We want to find the measure of angle A to the nearest degree.

Step 1: Plug in the Values

We can plug the values of a, b, and c into the law of cosines formula:

c² = a² + b² - 2ab * cos(C)

We know that c = 4.7 cm, a = 2.6 cm, and b = 4.3 cm. We can plug these values into the formula:

(4.7)² = (2.6)² + (4.3)² - 2(2.6)(4.3) * cos(A)

Step 2: Simplify the Equation

We can simplify the equation by evaluating the squares and the product:

21.49 = 6.76 + 18.49 - 22.28 * cos(A)

Step 3: Isolate the Cosine Term

We can isolate the cosine term by subtracting the constants from both sides of the equation:

21.49 - 6.76 - 18.49 = -22.28 * cos(A)

Step 4: Simplify the Constants

We can simplify the constants by combining them:

-3.76 = -22.28 * cos(A)

Step 5: Divide Both Sides by -22.28

We can divide both sides of the equation by -22.28 to isolate the cosine term:

cos(A) = -3.76 / -22.28

Step 6: Evaluate the Cosine

We can evaluate the cosine by dividing the numerator by the denominator:

cos(A) = 0.168

Step 7: Find the Measure of Angle A

We can find the measure of angle A by taking the inverse cosine of both sides of the equation:

A = arccos(0.168)

Step 8: Evaluate the Inverse Cosine

We can evaluate the inverse cosine by using a calculator or a trigonometric table:

A ≈ 80.5°

Conclusion

In this article, we used the law of cosines to solve a problem involving a triangle with known side lengths. We plugged the values into the formula, simplified the equation, isolated the cosine term, and evaluated the inverse cosine to find the measure of angle A to the nearest degree.

The Law of Cosines Formula

The law of cosines formula is:

c² = a² + b² - 2ab * cos(C)

where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.

Applications of the Law of Cosines

The law of cosines has many applications in mathematics and science, including:

  • Geometry: The law of cosines is used to find the length of a side of a triangle when we know the lengths of the other two sides and the measure of the angle between them.
  • Trigonometry: The law of cosines is used to find the measure of an angle in a triangle when we know the lengths of the three sides.
  • Physics: The law of cosines is used to find the distance between two points in a plane when we know the coordinates of the points.
  • Engineering: The law of cosines is used to find the length of a side of a triangle when we know the lengths of the other two sides and the measure of the angle between them.

Real-World Examples of the Law of Cosines

The law of cosines has many real-world applications, including:

  • Surveying: The law of cosines is used to find the distance between two points on the Earth's surface when we know the coordinates of the points.
  • Navigation: The law of cosines is used to find the distance between two points in a plane when we know the coordinates of the points.
  • Architecture: The law of cosines is used to find the length of a side of a triangle when we know the lengths of the other two sides and the measure of the angle between them.

Conclusion

In conclusion, the law of cosines is a powerful tool for solving triangles and finding the length of a side when we know the lengths of the other two sides and the measure of the angle between them. It has many applications in mathematics and science, and is used in a variety of real-world situations.