Answer: x ≈ 24.7
Step-by-step explanation: To solve for x in a right triangle, we can use the trigonometric ratios of sine, cosine and tangent. These ratios relate the lengths of the sides of a right triangle to the angles opposite them.
In this case, we are given the length of the long side (not the hypotenuse) of the right triangle, which is 42, and the angle between the hypotenuse and this side, which is 64 degrees. We can label this side as adjacent to the angle, and the unknown side x as opposite to the angle.
The trigonometric ratio that relates the opposite and adjacent sides of a right triangle is tangent. The formula for tangent is:
tan(angle) = opposite / adjacent
Plugging in the given values, we get:
tan(64) = x / 42
To solve for x, we need to multiply both sides by 42:
42 * tan(64) = x
Using a calculator, we can find that:
x ≈ 24.7
Therefore, the length of the short side of the right triangle is approximately 24.7 units.
We can check our answer by using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula for the Pythagorean theorem is:
a² + b² = c²
where a and b are the lengths of the legs (short and long sides) of the right triangle, and c is the length of the hypotenuse.
Plugging in our values, we get:
24.7² + 42² = c²
Simplifying, we get:
610.09 + 1764 = c²
Adding, we get:
2374.09 = c²
Taking the square root of both sides, we get:
c ≈ 48.7
This is a reasonable value for the hypotenuse of a right triangle with a long side of 42 and an angle of 64 degrees.
Therefore, our answer is correct. Hope this helps, and have a great day!