Solving Triangles

Solving triangles is an essential part of trigonometry, a branch of mathematics that deals with the study of relationships between side lengths and angles of triangles. The subject focuses on finding the unknown parts of a triangle when some known quantities are given. Triangles can be of different types such as right-angled, obtuse-angled or acute-angled, and solving them involves using trigonometric rules and laws.

Types of triangles

Before solving triangles, it is important to understand the basic types and characteristics of triangles:

• Right triangle: It has one angle of 90 degrees.
• Acute triangle: All angles are less than 90 degrees.
• Obtuse triangle: It has one angle more than 90 degrees.
• Scalene triangle: All sides have different lengths.
• Isosceles triangle: Two sides are of equal length.
• Equilateral triangle: All three sides are of equal length.

Solving right triangles

A right triangle is simplest to solve using the basic trigonometric functions and the Pythagorean Theorem.

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

c² = a² + b²

where c is the hypotenuse, and a and b are the other two sides.

Example of the Pythagorean Theorem

Suppose you have a right triangle with one side 3 units and the other side 4 units. Find the length of the hypotenuse.

c² = 3² + 4² 
c² = 9 + 16 
c² = 25 
c = √25 
c = 5

Thus, the hypotenuse of the triangle is 5 units.

Trigonometric ratios

The elementary trigonometric ratios used in a right-angled triangle are:

• Sine (sin): opposite side / hypotenuse
• Cosine (cos): adjacent side / hypotenuse
• Tangent (tan): Opposite side/adjacent side

Example of trigonometric ratios

Given: The angle between the base and the hypotenuse is β.

sin(β) = Opposite / Hypotenuse = 3 / 5 
cos(β) = Adjacent / Hypotenuse = 4 / 5 
tan(β) = Opposite / Adjacent = 3 / 4

Solving non-concurrent triangles

For non-right triangles, we use other rules, such as the Law of Sines and the Law of Cosines.

Law of Sines

This rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite it remains constant.

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the lengths of the sides, and A, B, and C are opposite angles.

Example of the Law of Sines

Given: a = 7, A = 30°, b = 10 Find angle B

a/sin(A) = b/sin(B) 
7/sin(30°) = 10/sin(B) 
sin(B) = 10 * sin(30°) / 7

The value of sin(30°) is 0.5, so:

sin(B) = 10 * 0.5 / 7 
sin(B) = 5 / 7 ≈ 0.714 
B = arcsin(0.714) ≈ 45.57°

Law of Cosines

This rule is useful when we know two sides of a triangle and the angle between them and we have to find the third side.

c² = a² + b² - 2ab * cos(C)

Example of the Law of Cosines

Find the third side of a triangle where a = 8 , b = 6, and C = 60°.

c² = a² + b² - 2ab * cos(C) 
c² = 8² + 6² - 2 * 8 * 6 * cos(60°) 
c² = 64 + 36 - 48 * 0.5 
c² = 64 + 36 - 24 
c² = 76 
c = √76 ≈ 8.72

Conclusion

Solving triangles requires understanding and applying various trigonometric rules and laws. Right triangles use simple rules like the Pythagorean Theorem and basic trigonometric ratios. Non-right triangles require more advanced rules like the Law of Sines and the Law of Cosines. Practicing different scenarios will help develop a deeper understanding of trigonometry and solving triangles.

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