Determine which of the two acute angles has a cosine of 4/5.
Understand the Problem
The question is asking to determine which of the two acute angles in a right triangle has a cosine value of \frac{4}{5}. This involves applying the properties of cosine in right triangles based on the given side lengths.
Answer
Angle \( A \) has a cosine of \( \frac{4}{5} \).
Answer for screen readers
Angle ( A ) has a cosine value of ( \frac{4}{5} ).
Steps to Solve
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Identify the triangle's sides The sides of the triangle are given:
- Opposite side to angle ( A ) is ( 6 ) (length ( BC )).
- Adjacent side to angle ( A ) is ( 8 ) (length ( AC )).
- Hypotenuse is ( 10 ) (length ( AB )).
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Apply the cosine definition The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. Therefore, for angle ( A ): $$ \cos(A) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{8}{10} = \frac{4}{5} $$
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Determine the angle Since ( \cos(A) = \frac{4}{5} ), we conclude that angle ( A ) has a cosine value of ( \frac{4}{5} ).
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Find the other angle's cosine The other acute angle ( C ) can be determined using: $$ \cos(C) = \frac{BC}{AB} = \frac{6}{10} = \frac{3}{5} $$
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Summarize the angles
- Angle ( A ) has a cosine of ( \frac{4}{5} ).
- Angle ( C ) has a cosine of ( \frac{3}{5} ).
Angle ( A ) has a cosine value of ( \frac{4}{5} ).
More Information
In a right triangle, the definition of cosine relates strictly to the lengths of the sides. This triangle has an angle ( A ) opposite the side of length ( 6 ) and adjacent to a side of length ( 8 ), giving it the stated cosine value.
Tips
- Confusing the opposite and adjacent sides can lead to errors in calculating the cosine. Always double-check the sides relevant to the angle being evaluated.
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