📐 Level 2 - Topic 5: Law of Cosines - Solving Triangles with SAS & SSS Cases 💪△

1) Completing the Triangle Toolkit - When Law of Sines Isn't Enough 🛠️

In the previous topic, we mastered the Law of Sines, a powerful tool for solving oblique triangles in ASA, AAS, and SSA cases. However, there are situations where the Law of Sines is not directly applicable, specifically when we are given:

  • Side-Angle-Side (SAS): Two sides and the included angle (the angle between them).
  • Side-Side-Side (SSS): All three sides of the triangle.

For these cases, we need another tool in our trigonometric toolkit – the Law of Cosines. The Law of Cosines is a generalization of the Pythagorean theorem and is essential for solving triangles in SAS and SSS scenarios, and it also works for all types of triangles, including right-angled triangles!

Why the Law of Cosines?

  • Solving SAS and SSS Triangles: Law of Cosines allows us to solve triangles when we know two sides and the included angle (SAS) or all three sides (SSS).
  • Generalization of Pythagorean Theorem: For right triangles, the Law of Cosines reduces to the Pythagorean theorem. It extends this relationship to all triangles.
  • Versatile Applications: Essential in various fields, including engineering, physics, surveying, and navigation, for problems involving triangle calculations.

In this topic, we will explore the Law of Cosines, understand its different forms, learn how to apply it to solve SAS and SSS triangle problems, and see how it relates to the Pythagorean theorem.

2) The Law of Cosines - Relating Sides and Angles with Cosine 📐🔗

The Law of Cosines provides a relationship between the sides of a triangle and the cosine of one of its angles. Again, consider a triangle \( ABC \) with angles \( A, B, C \) and opposite sides \( a, b, c \).

Definition: Law of Cosines (Standard Forms)

In any triangle \( ABC \), the square of a side is equal to the sum of the squares of the other two sides minus twice the product of these two sides and the cosine of the angle included between them. Mathematically, this can be expressed in three forms:

\( a^2 = b^2 + c^2 - 2bc \cos(A) \)

\( b^2 = a^2 + c^2 - 2ac \cos(B) \)

\( c^2 = a^2 + b^2 - 2ab \cos(C) \)

Understanding the Formula:

  • Each formula relates one side of the triangle to the cosine of its opposite angle and the other two sides. For example, the first formula \( a^2 = b^2 + c^2 - 2bc \cos(A) \) relates side \( a \) to angle \( A \) and sides \( b \) and \( c \).
  • Notice the similarity to the Pythagorean theorem: \( a^2 = b^2 + c^2 \) (if \( \angle A = 90^\circ \), then \( \cos(A) = \cos(90^\circ) = 0 \), and the term \( -2bc \cos(A) \) becomes zero). Thus, the Law of Cosines is a generalization of the Pythagorean theorem.
  • To use the Law of Cosines, you'll typically use the formula that involves the known sides and the angle you know or want to find.

3) When to Use the Law of Cosines - Identifying Applicable Cases 🔍

The Law of Cosines is primarily used in the following cases when solving triangles:

  • Case 1: Side-Angle-Side (SAS) - We know two sides and the included angle. For example, we know sides \( b, c \) and angle \( A \). We can use the Law of Cosines to find the side opposite to the known angle (side \( a \)).
  • Case 2: Side-Side-Side (SSS) - We know all three sides of the triangle. For example, we know sides \( a, b, c \). We can use the Law of Cosines to find any of the angles (e.g., angle \( A \), \( B \), or \( C \)).

Cases Where Law of Cosines Can Also Be Used (but Law of Sines might be simpler for some):

While Law of Cosines is essential for SAS and SSS, it can also be used in other cases, although Law of Sines might be more straightforward in some situations:

  • ASA or AAS Cases: If you know two angles and a side, you can technically use Law of Cosines, but it's often more direct to use Law of Sines or simply find the third angle using the angle sum property and then use Law of Sines.
  • SSA Case (Ambiguous): Law of Cosines can be used to analyze the SSA case as well, though the quadratic equation resulting from it might make it less direct than using Law of Sines for initially finding possible angles. However, Law of Cosines provides a different perspective on the SSA ambiguity.

4) Solving Triangles - Examples Using the Law of Cosines 🚀

Example 1: SAS Case - Finding the Unknown Side

In triangle \( ABC \), \( b = 8 \) cm, \( c = 5 \) cm, and \( \angle A = 60^\circ \). Find side \( a \).

  1. Apply Law of Cosines to find \( a \): Use the formula \( a^2 = b^2 + c^2 - 2bc \cos(A) \).
  2. Substitute the known values: \( a^2 = 8^2 + 5^2 - 2 \times 8 \times 5 \times \cos(60^\circ) \).
  3. Calculate: \( a^2 = 64 + 25 - 80 \times \frac{1}{2} = 89 - 40 = 49 \).
  4. Solve for \( a \): \( a = \sqrt{49} = 7 \) cm. (Since side length must be positive, we take the positive square root).

Solution: Side \( a = 7 \) cm.

Example 2: SSS Case - Finding an Unknown Angle

In triangle \( ABC \), \( a = 7 \) units, \( b = 8 \) units, and \( c = 13 \) units. Find angle \( C \).

  1. Use the Law of Cosines formula to solve for \( \cos(C) \): Start with \( c^2 = a^2 + b^2 - 2ab \cos(C) \). Rearrange to solve for \( \cos(C) \):

    \( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \)

  2. Substitute the known values: \( \cos(C) = \frac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8} = \frac{49 + 64 - 169}{112} = \frac{-56}{112} = -\frac{1}{2} \).
  3. Find angle \( C \): \( C = \arccos(-\frac{1}{2}) \). Since cosine is \( -\frac{1}{2} \), and we know for arccosine range is \( [0^\circ, 180^\circ] \), \( C = 120^\circ \).

Solution: Angle \( C = 120^\circ \).

Example 3: SSS Case - Finding All Angles

In triangle \( ABC \), \( a = 5 \), \( b = 7 \), and \( c = 8 \). Find all angles \( A, B, C \).

  1. Find angle \( A \): Use the Law of Cosines formula solved for \( \cos(A) \): \( \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{7^2 + 8^2 - 5^2}{2 \times 7 \times 8} = \frac{49 + 64 - 25}{112} = \frac{88}{112} = \frac{11}{14} \). \( A = \arccos(\frac{11}{14}) \approx 38.21^\circ \).
  2. Find angle \( B \): Use the Law of Cosines formula solved for \( \cos(B) \): \( \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} = \frac{5^2 + 8^2 - 7^2}{2 \times 5 \times 8} = \frac{25 + 64 - 49}{80} = \frac{40}{80} = \frac{1}{2} \). \( B = \arccos(\frac{1}{2}) = 60^\circ \).
  3. Find angle \( C \): We can use the angle sum property: \( C = 180^\circ - A - B \approx 180^\circ - 38.21^\circ - 60^\circ = 81.79^\circ \). (Alternatively, you could use Law of Cosines to find \( C \) and verify).

Solution: \( \angle A \approx 38.21^\circ \), \( \angle B = 60^\circ \), \( \angle C \approx 81.79^\circ \).

5) Practice Questions 🎯

5.1 Fundamental – Applying Law of Cosines (SAS & SSS Cases)

1. In triangle \( ABC \), \( b = 10 \) m, \( c = 12 \) m, and \( \angle A = 48^\circ \). Find side \( a \).

2. In triangle \( XYZ \), \( x = 9 \) cm, \( z = 14 \) cm, and \( \angle Y = 70^\circ \). Find side \( y \).

3. In triangle \( PQR \), \( p = 6 \) units, \( q = 8 \) units, and \( \angle R = 90^\circ \). Find side \( r \) using Law of Cosines and verify using Pythagorean theorem.

4. In triangle \( DEF \), \( d = 15 \) inches, \( e = 18 \) inches, and \( f = 20 \) inches. Find angle \( F \).

5. In triangle \( KLM \), \( k = 25 \) cm, \( l = 30 \) cm, and \( m = 40 \) cm. Find angle \( K \).

6. Triangle \( ABC \) has sides \( a = 11 \), \( b = 13 \), \( c = 17 \). Find angle \( B \).

7. Given triangle \( RST \) with \( r = 5 \), \( s = 5 \), and \( \angle T = 120^\circ \). Find side \( t \).

8. In triangle \( UVW \), \( u = 16 \), \( v = 16 \), \( w = 16 \). Find all angles \( U, V, W \) using Law of Cosines. (What type of triangle is this?)

9. In triangle \( ABC \), \( b = 20 \), \( c = 20 \), and \( \angle A = 90^\circ \). Find side \( a \) using Law of Cosines and verify using Pythagorean theorem. (What type of triangle is this?)

10. Triangle \( XYZ \) has sides \( x = 3 \), \( y = 4 \), \( z = 5 \). Find angle \( Z \). (What type of triangle is this?)

11. Triangle \( ABC \) has \( a=14 \), \( c=9 \) and angle \( B = 52^\circ \). Find side \( b \).

12. Triangle \( PQR \) has sides \( p=15 \), \( q=20 \), \( r=25 \). Find angle \( Q \).

5.2 Challenging – Problem Solving and Applications 💪🚀

1. Two airplanes leave an airport at the same time. Airplane 1 flies in a direction of N70°E at 400 mph, and Airplane 2 flies in a direction of S20°E at 500 mph. How far apart are the airplanes after 2 hours?

2. A triangular plot of land has sides of lengths 150 meters, 200 meters, and 250 meters. Find the largest angle in this plot of land.

3. A baseball diamond is a square with sides of 90 feet. The pitcher's mound is 60.5 feet from home plate. How far is the pitcher's mound from first base?

4. A parallelogram has sides of length 8 cm and 12 cm, and an angle of \( 75^\circ \) between them. Find the length of the longer diagonal of the parallelogram. (Hint: Consider the triangle formed by the sides and the diagonal).

5. (Conceptual) Explain why the Law of Cosines is considered a generalization of the Pythagorean theorem. Under what condition does the Law of Cosines simplify to the Pythagorean theorem? Show mathematically.

6) Summary - Mastering the Law of Cosines for SAS & SSS Triangles 🎉

  • Law of Cosines: Provides formulas to relate sides and angles in any triangle: \( a^2 = b^2 + c^2 - 2bc \cos(A) \) (and cyclic variations).
  • Solving SAS Triangles: Used to find the third side when two sides and the included angle are known.
  • Solving SSS Triangles: Used to find any angle when all three sides are known. Can be rearranged to \( \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \) (and cyclic variations).
  • Generalization of Pythagorean Theorem: Law of Cosines becomes Pythagorean theorem when the angle is \( 90^\circ \) (cosine of \( 90^\circ \) is 0).
  • Applicable Cases: Primarily used for SAS and SSS cases, where Law of Sines is not directly applicable.
  • Toolkit Completion: With Law of Sines and Law of Cosines, we now have powerful tools to solve a wide range of triangle problems.

Excellent work! You have now mastered the Law of Cosines, adding another powerful tool to your trigonometry arsenal. Understanding both the Law of Sines and Law of Cosines gives you the ability to solve virtually any triangle problem given sufficient information. In the following topics, we will build upon these triangle-solving skills and explore more advanced trigonometric identities and applications. Keep practicing and you'll become a true trigonometry master! 📐💪△🌟

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