In triangle ABC, m∠B = 90°, cos(C) = 16/17, and BC = 30 units.
Based on this information, m∠A = ___°, m∠C = ___°, and AB = ___ units.
Note that the angle measures are rounded to the nearest degree.
Hint: First solve for m∠C by using the cosine equation given.
Answer (2)
To solve the problem, we need to find the measures of angles m ∠ A and m ∠ C , and the length of A B in triangle △ A BC with the given information. The triangle is a right triangle with m ∠ B = 9 0 ∘ .
Find m ∠ C :
The cosine of angle C is given as cos ( C ) = 17 16 . Using the cosine definition: cos ( C ) = h y p o t e n u se a d ja ce n t In triangle △ A BC , with respect to angle C , BC is the adjacent side and A B is the hypotenuse. So, the formula becomes:
A B BC = 17 16 Given BC = 30 units: A B 30 = 17 16 Solving this for A B :
A B = 16 30 × 17 = 31.875
Find m ∠ A :
Since △ A BC is a right triangle, the sum of angles in a triangle is 18 0 ∘ :
m ∠ A + m ∠ B + m ∠ C = 18 0 ∘ Knowing m ∠ B = 9 0 ∘ and m ∠ C ≈ cos − 1 ( 17 16 ) ≈ 1 8 ∘ , we find m ∠ A :
m ∠ A + 9 0 ∘ + 1 8 ∘ = 18 0 ∘ m ∠ A ≈ 7 2 ∘
Rounding and summarizing the answers :
m ∠ A ≈ 7 2 ∘
m ∠ C ≈ 1 8 ∘
A B ≈ 32 (rounded to the nearest whole number)
Therefore, the answer is m ∠ A = 7 2 ∘ , m ∠ C = 1 8 ∘ , and A B = 32 units.
In triangle ABC, the measures are m∠A = 72°, m∠C = 18°, and the length of AB is approximately 32 units when rounded to the nearest whole number.
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