In a figure angle a =90, AB=16cm and tan angle C=4/5.find(a)AC ,(b)BC(C) ANGLE C(D)DISTANCE FROM A TO BC.
To solve this problem, we will use trigonometric concepts and the Pythagorean theorem.
Given:
- In a figure, angle a is 90 degrees.
- Side AB has a length of 16 cm.
- The tangent of angle C is 4/5.
Let's solve for each part of the problem:
(a) Finding AC:
Using the Pythagorean theorem, we can find the length of side AC.
According to the Pythagorean theorem:
AC^2 = AB^2 - BC^2
Since angle a is 90 degrees, triangle ABC is a right triangle.
Substituting the known values:
AC^2 = 16^2 - BC^2
(b) Finding BC:
We know that the tangent of angle C is defined as the ratio of the opposite side (BC) to the adjacent side (AB).
tan(C) = BC / AB
Substituting the known values:
4/5 = BC / 16
To find BC, we can cross-multiply and solve for BC:
5 * BC = 4 * 16
BC = 4 * 16 / 5
BC = 64 / 5
(c) Finding angle C:
Using the inverse tangent function, we can find angle C.
tan(C) = 4/5
C = atan(4/5)
(d) Finding the distance from A to BC:
Since angle a is 90 degrees and triangle ABC is a right triangle, the distance from A to BC is the same as the length of side AC.
Now you can use a calculator to find the values of AC and angle C.
AC = √(16^2 - BC^2)
C = atan(4/5)