Teach me some laws of trigonometry s.5 and how to apply them
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Answer
Trigonometry has several laws that help us solve triangles. The two most fundamental are the Law of Sines and the Law of Cosines.
Step-by-step solution
6 stepsLaw of Sines
This law relates the lengths of the sides of a triangle to the sines of its opposite angles. For a triangle with sides a, b, c and opposite angles A, B, C, the law states:
(a)/(sin A) = (b)/(sin B) = (c)/(sin C)
Application: The Law of Sines is useful when you know:
- Two angles and one side (AAS or ASA).
- Two sides and an angle opposite one of them (SSA — this case can sometimes have two possible solutions, known as the ambiguous case).
Example:
Consider a triangle where angle A = 30°, angle B = 45°, and side a = 10. We can find side b using the Law of Sines:
Set up the proportion
(a)/(sin A) = (b)/(sin B)
(10)/(sin 30°) = (b)/(sin 45°)Substitute known sine values (sin 30° = 0.5, sin 45° = √(2)/2 ≈ 0.707)
(10)/(0.5) = (b)/(√(2)/2)
Solve for b
20 = (b)/(√(2)/2)
b = 20 × (√(2)/2)
b = 10√2Law of Cosines
This law is an extension of the Pythagorean theorem and relates the lengths of the sides of a triangle to the cosine of one of its angles. It states:c² = a² + b² - 2ab cos C
a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos BApplication: The Law of Cosines is useful when you know:
- All three sides (SSS).
- Two sides and the included angle (SAS).
Example:
Consider a triangle where side a = 7, side b = 8, and angle C = 60°. We can find side c using the Law of Cosines:Use the formula
c² = a² + b² - 2ab cos C
c² = 7² + 8² - 2(7)(8) cos 60°Substitute known values (cos 60° = 0.5)
c² = 49 + 64 - 2(56)(0.5)
c² = 113 - 56Solve for c
c² = 57
c = √57
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