Answer:
x = 21.8°
y = 4
z = 10
Note that 4 and 10 are approximations, which are usually acceptable answers when you have multiple digits in the decimal place.
If your teacher usually desires a more exact answer, you can use:
y = 3.999631589
z = 9.999792356
Explanation:
Finding x:
The Triangle Sum Theorem says that the sum of a triangle's interior angles equals 180°,
Thus, we can find x by subtracting the sum of the 68.2° angle and the 90° angle from 180:
x + 68.2 + 90 = 180
x = 180 - (68.2 + 90)
x = 180 - 158.2
x = 21.8
Thus, x = 21.8°.
Finding y:
The 90° angle is called a right angle, meaning this is a right triangle.
This allows to find y by using one of the trigonometric ratios.
We see that when the 68.2° angle is the reference angle, the side that is y units is the adjacent side and the side that is 10.77 units is the hypotenuse.
Thus, we can find y using the cosine ratio, whose general equation is given by:
cos (θ) = adjacent / hypotenuse, where
- θ is the reference angle.
Since the side that is y units is the adjacent side, we can rewrite the equation using y:
cos (θ) = y / hypotenuse
Thus, we can plug in 68.2 for θ and 10.77 for the hypotenuse to find y, the length of the adjacent side:
(cos (68.2) = y / 10.77)
10.77 * cos (68.2) = y
3.999631589 = y
4.0 = y
Thus, y is about 4 units.
If you want to use an exact answer, you can write y = 3.999631589.
Finding z:
In a right triangle, the sum of the squares shortest sides called legs is equal to the square of the longest side called the hypotenuse.
The formula is shown by the Pythagorean theorem, which is given by:
a^2 + b^2 = c^2, where
- a and b are the legs,
- and c is the hypotenuse.
We'll need to use to the exact answer for y to find z.
Thus, we can plug in 10.77 * cos (68.2) for a and 10.77 for c to find b (aka z in the triangle:
(10.77 * cos (68.2))^2 + b^2 = 10.77^2
b^2 = 10.77^2 - (10.77 * cos (68.2))^2
b = √(10.77^2 - (10.77 * cos (68.2))^2)
b = 9.999792356
b = 10
z = 10
Thus, z is approximately 10 units.
If you prefer to use the exact answer, write:
z = 9.999792356.