Given An Angle And Opposite Side In A Triangle, Find Constraints On The Side Adjacent To The Angle

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Introduction

In trigonometry, we often encounter problems involving triangles with known angles and side lengths. Given an angle and its opposite side in a triangle, we can use trigonometric ratios to find constraints on the side adjacent to the angle. In this article, we will explore the constraints on the side adjacent to an angle in a triangle, given an angle and its opposite side.

Problem Statement

In triangle ABCABC, we are given an angle A=42°A = 42°, and its opposite side length, a=38a = 38. We want to find the constraints on the side adjacent to angle AA, denoted as bb.

i) Unique Triangle

For a unique triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's denote the length of side bb as xx. We can write the following inequalities:

  • a+b>ca + b > c
  • a+c>ba + c > b
  • b+c>ab + c > a

Substituting the given values, we get:

  • 38+x>c38 + x > c
  • 38+c>x38 + c > x
  • x+c>38x + c > 38

We also know that the sum of the angles in a triangle is 180°180°. Since we are given angle A=42°A = 42°, we can find the other two angles:

  • B=180°−42°−CB = 180° - 42° - C
  • C=180°−42°−BC = 180° - 42° - B

Using the Law of Sines, we can relate the side lengths to the sines of the angles:

  • asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Substituting the given values, we get:

  • 38sin⁡42°=xsin⁡B=csin⁡C\frac{38}{\sin 42°} = \frac{x}{\sin B} = \frac{c}{\sin C}

We can simplify the equation by using the fact that sin⁡B=sin⁡(180°−42°−C)=sin⁡(138°−C)\sin B = \sin (180° - 42° - C) = \sin (138° - C):

  • 38sin⁡42°=xsin⁡(138°−C)=csin⁡C\frac{38}{\sin 42°} = \frac{x}{\sin (138° - C)} = \frac{c}{\sin C}

Now, we can use the fact that sin⁡(138°−C)=sin⁡Ccos⁡138°+cos⁡Csin⁡138°\sin (138° - C) = \sin C \cos 138° + \cos C \sin 138°:

  • 38sin⁡42°=xsin⁡Ccos⁡138°+cos⁡Csin⁡138°=csin⁡C\frac{38}{\sin 42°} = \frac{x}{\sin C \cos 138° + \cos C \sin 138°} = \frac{c}{\sin C}

Simplifying the equation, we get:

  • 38sin⁡Ccos⁡138°+38cos⁡Csin⁡138°=xsin⁡42°38 \sin C \cos 138° + 38 \cos C \sin 138° = x \sin 42°

Using the fact that sin⁡42°=sin⁡(180°−138°)=sin⁡42°\sin 42° = \sin (180° - 138°) = \sin 42°, we can simplify the equation:

  • 38sin⁡Ccos⁡138°+38cos⁡Csin⁡138°=xsin⁡42°38 \sin C \cos 138° + 38 \cos C \sin 138° = x \sin 42°

Now, we can use the fact that cos⁡138°=−cos⁡42°\cos 138° = -\cos 42° and sin⁡138°=sin⁡42°\sin 138° = \sin 42°:

  • −38sin⁡Ccos⁡42°+38cos⁡Csin⁡42°=xsin⁡42°-38 \sin C \cos 42° + 38 \cos C \sin 42° = x \sin 42°

Simplifying the equation, we get:

  • 38cos⁡Csin⁡42°=xsin⁡42°38 \cos C \sin 42° = x \sin 42°

Dividing both sides by sin⁡42°\sin 42°, we get:

  • 38cos⁡C=x38 \cos C = x

Now, we can use the fact that cos⁡C=cos⁡(180°−42°−B)=−cos⁡(42°+B)\cos C = \cos (180° - 42° - B) = -\cos (42° + B):

  • 38(−cos⁡(42°+B))=x38 (-\cos (42° + B)) = x

Simplifying the equation, we get:

  • −38cos⁡(42°+B)=x-38 \cos (42° + B) = x

Now, we can use the fact that cos⁡(42°+B)=cos⁡(42°)cos⁡B−sin⁡(42°)sin⁡B\cos (42° + B) = \cos (42°) \cos B - \sin (42°) \sin B:

  • −38(cos⁡(42°)cos⁡B−sin⁡(42°)sin⁡B)=x-38 (\cos (42°) \cos B - \sin (42°) \sin B) = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(180°−138°)=cos⁡42°\cos (42°) = \cos (180° - 138°) = \cos 42° and sin⁡(42°)=sin⁡42°\sin (42°) = \sin 42°:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Simplifying the equation, we get:

  • −38cos⁡(42°)cos⁡B+38sin⁡(42°)sin⁡B=x-38 \cos (42°) \cos B + 38 \sin (42°) \sin B = x

Now, we can use the fact that cos⁡(42°)=cos⁡(42°)\cos (42°) = \cos (42°) and sin⁡(42°)=sin⁡(42°)\sin (42°) = \sin (42°):

  • $-38 \cos (42°) \cos B + 38 \sin (42
    Constraints on the Side Adjacent to an Angle in a Triangle ===========================================================

Q&A

Q: What are the constraints on the side adjacent to an angle in a triangle? A: The constraints on the side adjacent to an angle in a triangle are given by the inequalities:

  • a+b>ca + b > c
  • a+c>ba + c > b
  • b+c>ab + c > a

Q: How do I find the constraints on the side adjacent to an angle in a triangle? A: To find the constraints on the side adjacent to an angle in a triangle, you can use the Law of Sines and the fact that the sum of the angles in a triangle is 180°180°. You can also use the fact that the cosine of an angle is equal to the adjacent side divided by the hypotenuse.

Q: What is the relationship between the side adjacent to an angle and the opposite side? A: The side adjacent to an angle and the opposite side are related by the cosine of the angle. Specifically, the cosine of the angle is equal to the adjacent side divided by the hypotenuse.

Q: How do I use the Law of Sines to find the constraints on the side adjacent to an angle in a triangle? A: To use the Law of Sines to find the constraints on the side adjacent to an angle in a triangle, you can start by writing the equation:

  • asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

You can then substitute the given values and simplify the equation to find the constraints on the side adjacent to the angle.

Q: What is the significance of the cosine of an angle in a triangle? A: The cosine of an angle in a triangle is significant because it is equal to the adjacent side divided by the hypotenuse. This means that the cosine of an angle can be used to find the length of the adjacent side.

Q: How do I use the fact that the sum of the angles in a triangle is 180°180° to find the constraints on the side adjacent to an angle in a triangle? A: To use the fact that the sum of the angles in a triangle is 180°180° to find the constraints on the side adjacent to an angle in a triangle, you can start by writing the equation:

  • A+B+C=180°A + B + C = 180°

You can then substitute the given values and simplify the equation to find the constraints on the side adjacent to the angle.

Q: What are some common mistakes to avoid when finding the constraints on the side adjacent to an angle in a triangle? A: Some common mistakes to avoid when finding the constraints on the side adjacent to an angle in a triangle include:

  • Failing to use the Law of Sines
  • Failing to use the fact that the sum of the angles in a triangle is 180°180°
  • Failing to simplify the equation correctly
  • Failing to check the constraints on the side adjacent to the angle

Q: How do I check the constraints on the side adjacent to an angle in a triangle? A: To check the constraints on the side adjacent to an angle in a triangle, you can use the inequalities:

  • a+b>ca + b > c
  • a+c>ba + c > b
  • b+c>ab + c > a

You can also use the fact that the cosine of an angle is equal to the adjacent side divided by the hypotenuse.

Q: What are some real-world applications of the constraints on the side adjacent to an angle in a triangle? A: Some real-world applications of the constraints on the side adjacent to an angle in a triangle include:

  • Architecture: The constraints on the side adjacent to an angle in a triangle are used to design buildings and bridges.
  • Engineering: The constraints on the side adjacent to an angle in a triangle are used to design machines and mechanisms.
  • Physics: The constraints on the side adjacent to an angle in a triangle are used to describe the motion of objects.

Conclusion

In conclusion, the constraints on the side adjacent to an angle in a triangle are given by the inequalities:

  • a+b>ca + b > c
  • a+c>ba + c > b
  • b+c>ab + c > a

These constraints can be used to find the length of the side adjacent to the angle, and they have many real-world applications in architecture, engineering, and physics.