In right ∆ABC, C is a right angle and sin A = . What is mB?

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To find the measure of angle B in right triangle ABC, we need to use the trigonometric relationship between the sine function and the angles in a right triangle.

Given that sin A = θ (let's say), we know that the sine of an angle is equal to the length of the side opposite to the angle divided by the length of the hypotenuse. In this case, sin A = BC / AC.

Since angle C is a right angle (90 degrees), we can use the Pythagorean theorem to relate the lengths of the sides of the triangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (AC^2) is equal to the sum of the squares of the other two sides (AB^2 + BC^2).

Now, let's substitute the given information into the equation. Since C is a right angle, we have AC^2 = AB^2 + BC^2.

Using the relationship sin A = BC / AC, we have:
θ = BC / AC.

Squaring both sides of the equation, we get:
θ^2 = (BC / AC)^2.

Next, replacing BC/AC with θ using the given information, we have:
θ^2 = θ^2.

This equation is always true since any real number squared is equal to itself squared.

Therefore, we cannot determine the measure of angle B (mB) based on the given information alone. More information is needed, such as the length of another side or angle measure.