How to calculate angle B if u are given AB=10,6 and AC=6 then angle D=alfa
To solve for angle B, we need to use the law of cosines and law of sines. Here's how you can do it step by step:
1. First, draw a triangle ABC with side lengths AB = 10,6 and AC = 6. Let angle B be angle β.
2. Apply the law of cosines to find angle B:
Cos(β) = (AB² + AC² - BC²) / (2 * AB * AC)
Substitute the given values: (10,6)² + (6)² - BC² = (2 * 10,6 * 6 * cos(β))
Simplify the equation and rearrange to solve for BC:
BC² = (10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))
BC = √[(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))]
3. Since we are given that angle D = α, and angle D and angle B are opposite sides in triangle ABC, we can conclude that angle D = angle B = β.
4. Use the law of sines to find the value of angle B:
Sin(β) / BC = Sin(α) / AC
Substitute the values: sin(β) / √[(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))] = sin(α) / 6
Simplify the equation and solve for sin(β):
sin(β) = (6 * sin(α)) / √[(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))]
Substitute sin(α) with sin(β):
sin(β) = (6 * sin(β)) / √[(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))]
Cross-multiply and simplify:
sin(β) * √[(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))] = 6 * sin(β)
√[(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β))] = 6
Square both sides and continue simplifying:
(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β)) = 6²
(10,6)² + (6)² - (2 * 10,6 * 6 * cos(β)) = 36
112,36 + 36 - (126,72 * cos(β)) = 36
148,36 - (126,72 * cos(β)) = 0
(126,72 * cos(β)) = 148,36
cos(β) = 148,36 / 126,72
cos(β) = 1,17
5. Now, use inverse cosine (arccos) to find angle β:
β = arccos(1,17)
Note: If the value of cos(β) exceeds 1 or is less than -1, it means that the given side lengths do not form a valid triangle.
Remember to use a calculator to compute the trigonometric functions and square roots accurately.