Step by step solution to "Find, correct to the nearest degree, the three..."

High School
Geometry
Trigonometry
Right triangles and trigonometry

FobelloE

Answered question

2021-06-07

Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
A(1,0,-1), B(5,-3,0), C(1,5,2)
Find

∠ C A B, ∠ A B C, ∠ B C A

Answer & Explanation

berggansS

Skilled2021-06-08Added 91 answers

$A B = (5, - 3, 0) - (1, 0, 1) = (4, - 3, 1)$
$A C = (1, 5, 2) - (1, 0, - 1) = (0, 5, 3)$
$| A B | = \sqrt{4^{2} + 3^{2} + 1^{2}} = \sqrt{26}$
$| A C | = \sqrt{0 + 5^{2} + 3^{2}} = \sqrt{34}$
Use:
$A B \cdot A C = | A B | \cdot | A C | \cos ∠ C A B$
$(4, - 3, 1) \cdot (0, 5, 3) = \sqrt{26} \cdot \sqrt{34} \cdot \cos ∠ C A B$
$4 \cdot 0 - 3 \cdot 5 + 1 \cdot 3 = \sqrt{26} \cdot \sqrt{34} \cdot \cos ∠ C A B$
$- 12 = \sqrt{26} \cdot \sqrt{34} \cdot \cos ∠ C A B$
$\cos ∠ C A B = - 0.404$
$∠ C A B = 113.8$ degree
$B A = (1, 0, - 1) - (5, - 3, 0) = (- 4, 3, - 1)$
$B C = (1, 5, 2) - (5, - 3, 0) = (- 4, 8, 2)$
$| B A | = \sqrt{26}$
$| B C | = \sqrt{4^{2} + 8^{2} + 2^{2}} = \sqrt{84}$
Use:
$B A \cdot B C = | B A | \cdot | B C | \cdot \cos ∠ A B C$
$(- 4, 3, - 1) \cdot (- 4, 8, 2) = \sqrt{26} \cdot \sqrt{84} \cdot \cos ∠ A B C$
$16 + 24 - 2 = \sqrt{26} \cdot \sqrt{84} \cdot \cos ∠ A B C$
$\cos ∠ A B C = 0.813$
$∠ A B C = 35.6$ degree
Sum of all $∠$ $= 180$
$∠ B C A + 35.6 + 113.8 = 180$
$∠ B C A = 30.6$ degree

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