What are the lengths of the legs of a right triangle in which one acute angle measures 19° and the hypotenuse is 15 units long?
15 cos19°
and
15 sin19°
Looks like it's time to review the basic trig functions.
To find the lengths of the legs of a right triangle, we can use trigonometric ratios. In this case, we are given the measure of one acute angle (19°) and the length of the hypotenuse (15 units).
Let's label the lengths of the legs as a and b. According to trigonometry, the sine, cosine, and tangent ratios are useful when dealing with right triangles.
Since we know the hypotenuse (15 units) and we want to find the lengths of the legs, it is appropriate to use the sine ratio.
The sine ratio is defined as the ratio of the length of the side opposite the angle to the hypotenuse.
sin(angle) = opposite/hypotenuse
For the given angle of 19°, we can use the sine ratio as follows:
sin(19°) = a/15
To solve for a, we can rearrange the equation:
a = sin(19°) * 15
Using a scientific calculator, we can find the value of sin(19°) and calculate the length of leg a:
a ≈ sin(19°) * 15 ≈ 5.03 units
Therefore, the length of one leg of the right triangle is approximately 5.03 units.
To find the other leg (b), we need to use the cosine ratio:
cos(angle) = adjacent/hypotenuse
cos(19°) = b/15
To solve for b, we rearrange the equation:
b = cos(19°) * 15
Using a scientific calculator, we can find the value of cos(19°) and calculate the length of leg b:
b ≈ cos(19°) * 15 ≈ 14.28 units
Therefore, the length of the other leg of the right triangle is approximately 14.28 units.