Question

A=
Pls help solve for A

Answer 1

a=3.75 un., b=11.25 un.

`x=7.5\ un.`

`y=(15√(3))/(4)\ un.`

`z=(15√(3))/(2)\ un.`

Explanation:

Given triangle is special 30°-60°-90° right triangle. The leg that is opposite to the angle of measure 30° is always equal to half of the hypotenuse. The hypotenuse is of length 15 units, the leg that is opposite to the 30° angle is leg with length of x units, then

`x=(15)/(2)=7.5\ un.`

In right triangle with hypotenuse x and legs y and a, angle opposite to the leg a is 30°, then

`a=(x)/(2)=(7.5)/(2)=3.75\ un.`

and

`b=15-a=15-3.75=11.25\ un.`

By the Pythagorean theorem,

`x^2=y^2+a^2,\\ \\7.5^2=3.75^2+y^2,\\ \\y^2=\left((15)/(2)\right)^2-\left((15)/(4)\right)^2=(225)/(4)-(225)/(16)=(675)/(16),\\ \\y=(15√(3))/(4)\ un.`

In right triangle with legs y and b and hypotenuse z, leg y is opposite to 30° angle, then

`z=2y=(15√(3))/(2)\ un.`

Answer 2

`a=3.75`

Explanation:

The hypotenuse of the large triangle is 15.

We can see that the side opposite of 30° angle is
`x`

Trigonometric ratio of SINE relates opposite and hypotenuse.

Thus we can write and cross multiply and solve:

`sin(A)=(Opposite)/(Hypotenuse)\\sin(30)=(x)/(15)\\x=15*sin(30)=7.5`

Now if you see the smallest triangle,
`a` is the adjacent side and
`x` becomes the hypotenuse of this triangle.

Trigonometric ratio of COSINE relates adjacent and hypotenuse.

Thus we can write and cross multiply and solve:

`cos(A)=(Adjacent)/(Hypotenuse)\\cos(60)=(a)/(7.5)\\a=7.5*cos(60)=3.75`

Thus
`a=3.75`