In a triangle with side lengths 7, 24, and 25, what is sin of the smallest angle?

Rationale

Sin of the smallest angle in the triangle is 7/25.

In a triangle with sides 7, 24, and 25, the smallest angle corresponds to the side with length 7. By using the sine rule, we can find that sin of this smallest angle is 7/25, which clearly identifies it as a specific value derived from the relationship between the side lengths and the triangle's angles.

A (7/25) CORRECT

This is the correct answer. In a 7–24–25 right triangle, the smallest angle is opposite the shortest side (7). Since the triangle is right-angled with hypotenuse 25, we use
sin(θ)= opposite /hypotenuse = 7/25

Thus, this choice correctly represents the sine of the smallest angle.

B (24/25) YOUR ANSWER

This choice suggests the sine of an angle opposite the side of length 24. However, since the side of length 7 is the shortest, it corresponds to the smallest angle, making this option incorrect. The sine of the angle opposite to the longest side (25) would be larger than 24/25, and hence, this value does not represent the sine of the smallest angle.

C (25/24) YOUR ANSWER

This option is greater than 1, which is not possible for the sine of any angle. The sine function is defined within the interval [0, 1], thus rendering this choice invalid for any angle in a triangle, let alone the smallest angle.

D (24/7) YOUR ANSWER

This value is not valid as a sine of any angle because it is greater than 1, and sine values must lie in the interval [0,1]. Therefore, 24/7 cannot represent the sine of any angle in a triangle and is incorrect in this context.

Conclusion

In conclusion, the sine of the smallest angle in a triangle with side lengths 7, 24, and 25 is correctly identified as 7/25, derived from the smallest side length. All other choices either misrepresent the sine value or do not adhere to the properties of trigonometric functions, reinforcing the notion that careful analysis of angle-side relationships is critical in solving such problems.

In a triangle with side lengths 7, 24, and 25, what is sin of the smallest angle?

Sin of the smallest angle in the triangle is 7/25.

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