Question

Triangle XYZ has sides XY=3”, YZ =4”, and XZ=5”. If angle Y is a right angle, and side YZ is opposite angle X , what is the tan of angle C ?

Answer 1

Answer:

tan(X) = 4/3 and tan(Z) = 3/4

Step-by-step explanation:

* Lets revise the trigonometry functions

- In any right angle triangle:

# The side opposite to the right angle is called the hypotenuse

# The other two sides are called the legs of the right angle

* If the name of the triangle is ABC, where B is the right angle

∴ The hypotenuse is AC

∴ AB and BC are the legs of the right angle

- ∠A and ∠C are two acute angles

- For angle A

# sin(A) = opposite/hypotenuse

∵ The opposite to ∠A is BC

∵ The hypotenuse is AC

∴ sin(A) = BC/AC

# cos(A) = adjacent/hypotenuse

∵ The adjacent to ∠A is AB

∵ The hypotenuse is AC

∴ cos(A) = AB/AC  

# tan(A) = opposite/adjacent

∵ The opposite to ∠A is BC

∵ The adjacent to ∠A is AB

∴ tan(A) = BC/AB

* Now lets solve the problem

- In Δ XYZ

∵ m∠Y = 90°

∵ XY = 3 inches , YZ = 4 inches , XZ = 5 inches

∴ The hypotenuse is XZ

∴ The legs of the right angle are XY and YZ

* To find tan(X) use YZ as opposite to it and XY as adjacent to it

∵ tan(X) = YZ/XY ⇒ opposite/adjacent

∴ tan(X) = 4/3

* To find tan(Z) use YX as opposite to it and ZY as adjacent to it

∵ tan(Z) = YX/ZY ⇒ opposite/adjacent

∴ tan(Z) = 3/4

* tan(X) = 4/3 and tan(Z) = 3/4