Triangle \[$ \triangle ABC \$\] Has Side Lengths Of \[$ A = 16 \$\], \[$ B = 16\sqrt{3} \$\], And \[$ C = 32 \$\] Inches.Part A: Determine The Measure Of Angle \[$ A \$\]. (5 Points)Part B: Show How To Use The

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Introduction

In this article, we will explore how to determine the measure of an angle in a triangle when the side lengths are given. We will use the Law of Cosines to find the measure of angle A in triangle ABC.

The Law of Cosines

The Law of Cosines is a fundamental concept in trigonometry that relates the side lengths of a triangle to the cosine of one of its angles. The Law of Cosines states that for any triangle with side lengths a, b, and c, and angle C opposite side c, the following equation holds:

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

Applying the Law of Cosines to Triangle ABC

We are given the side lengths of triangle ABC as a = 16, b = 16√3, and c = 32 inches. We want to find the measure of angle A. To do this, we will use the Law of Cosines to solve for cos A.

First, we will rearrange the Law of Cosines formula to isolate cos A:

cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}

Now, we will plug in the given side lengths:

cos⁡A=(163)2+(32)2−(16)22(163)(32)\cos A = \frac{(16\sqrt{3})^2 + (32)^2 - (16)^2}{2(16\sqrt{3})(32)}

Simplifying the expression, we get:

cos⁡A=768+1024−25610243\cos A = \frac{768 + 1024 - 256}{1024\sqrt{3}}

cos⁡A=153610243\cos A = \frac{1536}{1024\sqrt{3}}

cos⁡A=323\cos A = \frac{3}{2\sqrt{3}}

cos⁡A=32\cos A = \frac{\sqrt{3}}{2}

Finding the Measure of Angle A

Now that we have found cos A, we can use the inverse cosine function to find the measure of angle A.

A=cos⁡−1(32)A = \cos^{-1} \left(\frac{\sqrt{3}}{2}\right)

Using a calculator or trigonometric table, we find that:

A=60∘A = 60^{\circ}

Therefore, the measure of angle A is 60 degrees.

Conclusion

In this article, we used the Law of Cosines to find the measure of angle A in triangle ABC. We first rearranged the Law of Cosines formula to isolate cos A, then plugged in the given side lengths to solve for cos A. Finally, we used the inverse cosine function to find the measure of angle A. The measure of angle A is 60 degrees.

Part B: Using the Law of Cosines to Find Angle B

To find angle B, we will use the Law of Cosines again. This time, we will solve for cos B.

cos⁡B=a2+c2−b22ac\cos B = \frac{a^2 + c^2 - b^2}{2ac}

Plugging in the given side lengths, we get:

cos⁡B=(16)2+(32)2−(163)22(16)(32)\cos B = \frac{(16)^2 + (32)^2 - (16\sqrt{3})^2}{2(16)(32)}

Simplifying the expression, we get:

cos⁡B=256+1024−7681024\cos B = \frac{256 + 1024 - 768}{1024}

cos⁡B=5121024\cos B = \frac{512}{1024}

cos⁡B=12\cos B = \frac{1}{2}

Finding the Measure of Angle B

Now that we have found cos B, we can use the inverse cosine function to find the measure of angle B.

B=cos⁡−1(12)B = \cos^{-1} \left(\frac{1}{2}\right)

Using a calculator or trigonometric table, we find that:

B=60∘B = 60^{\circ}

Therefore, the measure of angle B is also 60 degrees.

Part C: Using the Law of Cosines to Find Angle C

To find angle C, we will use the Law of Cosines again. This time, we will solve for cos C.

cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}

Plugging in the given side lengths, we get:

cos⁡C=(16)2+(163)2−(32)22(16)(163)\cos C = \frac{(16)^2 + (16\sqrt{3})^2 - (32)^2}{2(16)(16\sqrt{3})}

Simplifying the expression, we get:

cos⁡C=256+768−10245123\cos C = \frac{256 + 768 - 1024}{512\sqrt{3}}

cos⁡C=05123\cos C = \frac{0}{512\sqrt{3}}

cos⁡C=0\cos C = 0

Finding the Measure of Angle C

Now that we have found cos C, we can use the inverse cosine function to find the measure of angle C.

C=cos⁡−1(0)C = \cos^{-1} (0)

Using a calculator or trigonometric table, we find that:

C=90∘C = 90^{\circ}

Therefore, the measure of angle C is 90 degrees.

Conclusion

Introduction

In our previous article, we explored how to use the Law of Cosines to find the measures of angles in a triangle when the side lengths are given. In this article, we will answer some frequently asked questions about solving triangles with the Law of Cosines.

Q: What is the Law of Cosines?

A: The Law of Cosines is a fundamental concept in trigonometry that relates the side lengths of a triangle to the cosine of one of its angles. It states that for any triangle with side lengths a, b, and c, and angle C opposite side c, the following equation holds:

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

Q: How do I use the Law of Cosines to find the measure of an angle?

A: To find the measure of an angle using the Law of Cosines, you need to rearrange the formula to isolate cos A, B, or C, then plug in the given side lengths to solve for cos A, B, or C. Finally, you can use the inverse cosine function to find the measure of the angle.

Q: What if I have two sides and the included angle? Can I use the Law of Cosines to find the third side?

A: Yes, you can use the Law of Cosines to find the third side of a triangle when you have two sides and the included angle. Simply rearrange the formula to isolate the third side, then plug in the given side lengths and the included angle to solve for the third side.

Q: What if I have two sides and the angle opposite one of them? Can I use the Law of Cosines to find the third side?

A: Yes, you can use the Law of Cosines to find the third side of a triangle when you have two sides and the angle opposite one of them. Simply rearrange the formula to isolate the third side, then plug in the given side lengths and the angle opposite one of them to solve for the third side.

Q: What if I have all three sides, but I don't know any of the angles? Can I use the Law of Cosines to find the angles?

A: Yes, you can use the Law of Cosines to find the angles of a triangle when you have all three sides. Simply rearrange the formula to isolate cos A, B, or C, then plug in the given side lengths to solve for cos A, B, or C. Finally, you can use the inverse cosine function to find the measure of the angles.

Q: What if I have a right triangle? Can I use the Law of Cosines to find the angles?

A: Yes, you can use the Law of Cosines to find the angles of a right triangle. However, you can also use the Pythagorean theorem to find the length of the hypotenuse, then use the inverse cosine function to find the measure of the angles.

Q: What are some common mistakes to avoid when using the Law of Cosines?

A: Some common mistakes to avoid when using the Law of Cosines include:

  • Not rearranging the formula correctly to isolate the desired side or angle
  • Not plugging in the correct side lengths or angle
  • Not using the correct inverse cosine function to find the measure of the angle
  • Not checking for errors in the calculations

Conclusion

In this article, we answered some frequently asked questions about solving triangles with the Law of Cosines. We hope that this article has been helpful in clarifying some of the common misconceptions and mistakes that people make when using the Law of Cosines. Remember to always double-check your calculations and use the correct inverse cosine function to find the measure of the angles.