Unit 12: Trigonometry Homework 6: Law Of Cosines This Is A 2-page Document!Task: Find Each Missing Measure. Round All Answers To The Nearest Tenth.Given Expression: $\[ 289 + 225 - 510 \cos(39) \\]Example Calculation: $\[ C^2 = 8^2 +
Introduction
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The Law of Cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. In this homework, we will apply the Law of Cosines to find missing measures in various triangles.
The Law of Cosines
The Law of Cosines states that for any triangle with sides of length a, b, and c, and angle C opposite side c, the following equation holds:
c² = a² + b² - 2ab * cos(C)
This equation allows us to find the length of side c, given the lengths of sides a and b, and the measure of angle C.
Example Calculation
Let's consider an example to illustrate how to use the Law of Cosines. Suppose we have a triangle with sides of length 8 and 15, and angle C measures 39°. We want to find the length of side c.
c² = 8² + 15² - 2(8)(15) * cos(39°) c² = 64 + 225 - 240 * cos(39°) c² = 289 - 240 * cos(39°)
Using a calculator, we can find the value of cos(39°) and substitute it into the equation:
c² = 289 - 240 * 0.777 c² = 289 - 186.48 c² = 102.52
Taking the square root of both sides, we get:
c ≈ √102.52 c ≈ 10.1
Therefore, the length of side c is approximately 10.1 units.
Homework Problems
Now, let's apply the Law of Cosines to find missing measures in various triangles.
Problem 1
In triangle ABC, side a measures 10 units, side b measures 12 units, and angle C measures 60°. Find the length of side c.
c² = a² + b² - 2ab * cos(C) c² = 10² + 12² - 2(10)(12) * cos(60°) c² = 100 + 144 - 240 * 0.5 c² = 244 - 120 c² = 124
c ≈ √124 c ≈ 11.1
Problem 2
In triangle DEF, side d measures 15 units, side e measures 20 units, and angle F measures 45°. Find the length of side f.
f² = d² + e² - 2de * cos(F) f² = 15² + 20² - 2(15)(20) * cos(45°) f² = 225 + 400 - 600 * 0.707 f² = 625 - 424.2 f² = 200.8
f ≈ √200.8 f ≈ 14.1
Problem 3
In triangle GHI, side g measures 8 units, side h measures 12 units, and angle I measures 30°. Find the length of side i.
i² = g² + h² - 2gh * cos(I) i² = 8² + 12² - 2(8)(12) * cos(30°) i² = 64 + 144 - 192 * 0.866 i² = 208 - 165.52 i² = 42.48
i ≈ √42.48 i ≈ 6.5
Problem 4
In triangle JKL, side j measures 10 units, side k measures 15 units, and angle L measures 60°. Find the length of side l.
l² = j² + k² - 2jk * cos(L) l² = 10² + 15² - 2(10)(15) * cos(60°) l² = 100 + 225 - 300 * 0.5 l² = 325 - 150 l² = 175
l ≈ √175 l ≈ 13.2
Problem 5
In triangle MNO, side m measures 12 units, side n measures 18 units, and angle O measures 45°. Find the length of side o.
o² = m² + n² - 2mn * cos(O) o² = 12² + 18² - 2(12)(18) * cos(45°) o² = 144 + 324 - 432 * 0.707 o² = 468 - 305.544 o² = 162.456
o ≈ √162.456 o ≈ 12.8
Problem 6
In triangle PQR, side p measures 15 units, side q measures 20 units, and angle R measures 30°. Find the length of side r.
r² = p² + q² - 2pq * cos(R) r² = 15² + 20² - 2(15)(20) * cos(30°) r² = 225 + 400 - 600 * 0.866 r² = 625 - 519.6 r² = 105.4
r ≈ √105.4 r ≈ 10.3
Problem 7
In triangle STU, side s measures 8 units, side t measures 12 units, and angle U measures 60°. Find the length of side u.
u² = s² + t² - 2st * cos(U) u² = 8² + 12² - 2(8)(12) * cos(60°) u² = 64 + 144 - 192 * 0.5 u² = 208 - 96 u² = 112
u ≈ √112 u ≈ 10.6
Problem 8
In triangle VWX, side v measures 10 units, side w measures 15 units, and angle X measures 45°. Find the length of side x.
x² = v² + w² - 2vw * cos(X) x² = 10² + 15² - 2(10)(15) * cos(45°) x² = 100 + 225 - 300 * 0.707 x² = 325 - 212.1 x² = 112.9
x ≈ √112.9 x ≈ 10.6
Problem 9
In triangle YZK, side y measures 12 units, side z measures 18 units, and angle K measures 60°. Find the length of side k.
k² = y² + z² - 2yz * cos(K) k² = 12² + 18² - 2(12)(18) * cos(60°) k² = 144 + 324 - 432 * 0.5 k² = 468 - 216 k² = 252
k ≈ √252 k ≈ 15.8
Problem 10
In triangle LMK, side l measures 15 units, side m measures 20 units, and angle K measures 45°. Find the length of side k.
k² = l² + m² - 2lm * cos(K) k² = 15² + 20² - 2(15)(20) * cos(45°) k² = 225 + 400 - 600 * 0.707 k² = 625 - 424.2 k² = 200.8
k ≈ √200.8 k ≈ 14.1
Conclusion
Introduction
In the previous article, we explored the Law of Cosines and applied it to find missing measures in various triangles. In this article, we will answer some frequently asked questions about the Law of Cosines and provide additional examples to help solidify your understanding of this important concept.
Q&A
Q: What is the Law of Cosines?
A: The Law of Cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It states that for any triangle with sides of length a, b, and c, and angle C opposite side c, the following equation holds:
c² = a² + b² - 2ab * cos(C)
Q: How do I use the Law of Cosines to find a missing side?
A: To use the Law of Cosines to find a missing side, you need to know the lengths of the other two sides and the measure of the angle opposite the missing side. You can then plug these values into the equation c² = a² + b² - 2ab * cos(C) and solve for c.
Q: What if I don't know the measure of the angle opposite the missing side?
A: If you don't know the measure of the angle opposite the missing side, you can use the Law of Sines to find it. The Law of Sines states that for any triangle with sides of length a, b, and c, and angles A, B, and C, the following equation holds:
a / sin(A) = b / sin(B) = c / sin(C)
You can use this equation to find the measure of the angle opposite the missing side.
Q: Can I use the Law of Cosines to find an angle?
A: Yes, you can use the Law of Cosines to find an angle. If you know the lengths of the sides of the triangle and the measure of one of the angles, you can use the equation c² = a² + b² - 2ab * cos(C) to find the measure of the angle opposite side c.
Q: What if I have a right triangle?
A: If you have a right triangle, you can use the Pythagorean Theorem to find the length of the missing side. The Pythagorean Theorem states that for any right triangle with sides of length a and b, and hypotenuse c, the following equation holds:
c² = a² + b²
You can use this equation to find the length of the missing side.
Q: Can I use the Law of Cosines to find the length of the hypotenuse of a right triangle?
A: Yes, you can use the Law of Cosines to find the length of the hypotenuse of a right triangle. If you know the lengths of the other two sides, you can use the equation c² = a² + b² - 2ab * cos(C) to find the length of the hypotenuse.
Additional Examples
Example 1
Find the length of side c in the triangle with sides of length 10 and 15, and angle C measures 60°.
c² = 10² + 15² - 2(10)(15) * cos(60°) c² = 100 + 225 - 300 * 0.5 c² = 325 - 150 c² = 175
c ≈ √175 c ≈ 13.2
Example 2
Find the length of side c in the triangle with sides of length 12 and 18, and angle C measures 45°.
c² = 12² + 18² - 2(12)(18) * cos(45°) c² = 144 + 324 - 432 * 0.707 c² = 468 - 305.544 c² = 162.456
c ≈ √162.456 c ≈ 12.8
Example 3
Find the length of side c in the triangle with sides of length 15 and 20, and angle C measures 30°.
c² = 15² + 20² - 2(15)(20) * cos(30°) c² = 225 + 400 - 600 * 0.866 c² = 625 - 519.6 c² = 105.4
c ≈ √105.4 c ≈ 10.3
Conclusion
In this article, we answered some frequently asked questions about the Law of Cosines and provided additional examples to help solidify your understanding of this important concept. We hope that this article has been helpful in your studies of trigonometry. If you have any further questions or need additional help, please don't hesitate to ask.