Can someone help me solve for which expression?

The base of the triangle is side #b = e#,

Please observe that all of the selections use the angle #D#; the correct equation for the height using the sine of angle #D# is:

#h = f sin(D)#

Substituting into the #A = 1/2bh# formula:

#A = 1/2e f sin(D)#

This is the fourth selection.

#sin D=h/f# so #h=fsinD#

#A=1/2xxbasexxheight#

#A=1/2xxexxfsinD#

#A=1/2efsinD#

Notice how all options involve #sinD#.

Remember the definition of #sin D = "opposite"/"hypotenuse"# for any right triangle in which #D < 90^@.#

In the small right triangle on the left, #h# is the "opposite" side to #D# and #f# is the hypotenuse. Thus, we have:

#sin D = h/f#

We can solve this for #h# by multiplying both sides by #f:#

#f sin D = cancelf * h/cancelf#

#=> color(red)h = color(red)(f sin D)#

We now have an expression for #h# in terms of #f# and #D.#

Using the classic formula for the area of a triangle #A= 1/2 bh,# where

• #b=e##" "# (side #e# is our base #b#)
• #h = f sin D##" "# (from above)

we can now replace #h# with its equivalent expression, as such:

#A = 1/2 ecolor(red)h#

becomes

#A = 1/2 ecolor(red)(fsinD)#