The answers are:
Question # (1): t = 15.8 (when rounded to the nearest tenth).
Question # (3): The length of Side EF is 7.5 (when rounded to the nearest tenth), and the length of side DF is 2.7 (when rounded to the nearest tenth).
Here is how to solve for "Question #1" and "Question #3":
Let us start with "Question #1":
Remember the phrase: "SOH CAH TOA"; in which:
Sin = opposite/hypotenuse; Cos = adjacent/ hypotenuse;
Tan = opposite/adjacent.
We are given an triangle; one angle has a measurement of 44°. Considering this given angle, the respective hypotenuse given is "22"—along with a variable "t" for the respective "adjacent" side.
Using the above: "SOH CAH TOA" method; it would make sense to use the "CAH" {cosine = adjacent/hypotenuse} part of the mnemonic to solve for "t".
Reason: We are given the angle measure 44°; the hypotenuse value ("22"); so the cosine of 44, or "cos 44" = adjacent/hypotenuse = "t" / 22 .
→ Write as: cos 44 = t / 22 ; Solve for "t" .
→ (cos 44)*22 = t ; → t = 22 * (cos 44);
→ t = 22 *(cos 44) = 22 * (0.719339800339) → Use a scientific calculator:
→ t = 15.825475607458 ; → Round to the nearest tenth, as indicated in the problem, to get:
→ t = 15.8 ; → which is the answer.
"Question #3: Find the missing lengths in the triangle. Round to the nearest tenth." [Note: There are 2 (two) missing lengths to find.].
To get the length of side "EF"; let us use the "CAH" part of the mnemonic,
"SOH CAH TOA", (in the aforementioned previous problem given).
The "CAH" stands for the formula/property to calculate for the "cosine"—specifically, "cosine = adjacent/hypotenuse". We choose this one because we are given the measure of "angle E = 20° "; and we are given the hypotenuse, "side ED = 8"; and we want to solve for the adjacent side, which is "EF" (which we will call: "x").
→ So, cos 20 = adjacent/ hypotenuse = x /8; → Solve for "x""
(cos 20)*8 = x ; → 8*(cos 20) = x ; →
→x = 8 * (cos 20) = 8 *(0.939692620786) → Use a scientific calculator: = 7.517540966288 ;→ Round to the nearest tenth, as indicated in the problem, to get:
→ 7.5 ; which is the answer for the length of side EF.
Now, we need to find the unknown length of the other side, "Side DF".
Here is one method:
Find the measure of "Angle D". Note that for any triangle, all three sides must add up to 180°. We know that this is a right triangle, with one angle ("Angle F") having a measurement of 90°. We are also given that another angle ("Angle E") has a measurement of 20°.
→ So, the measurement of "Angle D" = 180 - (90 + 20) = 180 - 110 = 70.
→ We want to solve for the length of side "DF", which is adjacent to "Angle D" (which we know is 70°).
Let us use the "CAH" part of the mnemonic,
"SOH CAH TOA", (in the aforementioned previous problem given).
The "CAH" stands for the formula/property to calculate for the "cosine"—specifically, "cosine = adjacent/hypotenuse". We choose this one because we are given the measure of angle "D"= 70°; and we are given the hypotenuse = 8. The "adjacent", or "Side DF", is the unknown value (let us make it "x").
So, cos 70 = adjacent / hypotenuse = x/ 8 ; Solve for "x" :
cos 70 = x / 8; (cos 70) * 8 = x ; 8 * (cos 70) = x;
x = 8 * (cos 70) = 8 * (0.342020143326) → Use a scientific calculator → = 2.736161146608 ;
→Round to the nearest tenth, as indicated in the problem, to get:
→ 2.7 ; which is our answer, for the length of side DF.
So, the answer for "Question 1" is: t = 15.8 (when rounded to the nearest tenth).
For "Question #3", the length of Side EF is 7.5 (when rounded to the nearest tenth), and the length of side DF is 2.7 (when rounded to the nearest tenth).
