in a right triangle XYZ where right angle at angle X, if z= 14 and angle Y = 27.7 degree. Find angle Z, side x and side y.
angle Z = 90- 27.7 = ....
y/14 = tan27.7°, y = 14tan27.7° = ...
14/x = cos27.7°, x = 14/cos27.7° = ...
To find angle Z, side x, and side y in a right triangle XYZ, where the right angle is at angle X, given z = 14 and angle Y = 27.7 degrees, we can use the trigonometric functions sine, cosine, and tangent.
1. Finding angle Z:
In a right triangle, the sum of the angles is always 180 degrees. Since the right angle at angle X is 90 degrees and angle Y is given as 27.7 degrees, we can calculate angle Z using:
Angle Z = 180 - 90 - 27.7
2. Finding side x (opposite angle X):
Using the sine function, we can find side x. The sine function relates the ratio between the length of the side opposite to an angle (in this case x) and the hypotenuse (in this case z). So, we have:
sin(X) = x / z
Substituting the values we have:
sin(X) = x / 14
Solving for x:
x = 14 * sin(X)
3. Finding side y (adjacent to angle X):
Using the cosine function, we can find side y. The cosine function relates the ratio between the length of the side adjacent to an angle (in this case y) and the hypotenuse (in this case z). So, we have:
cos(X) = y / z
Substituting the values we have:
cos(X) = y / 14
Solving for y:
y = 14 * cos(X)
Now, you can use a calculator to find the exact values of angle Z, side x, and side y by substituting the appropriate values into the formulas mentioned above.