Answer:
x = 13·568, y = 8·47 and z = 58
Explanation:
I don't think the question required you to use the Pythagoras theorem but here are some pointers anyway.
- Okay, the Pythagoras theorem is quite simple: a² + b² = c²
- Where c is always the hypotenuse, a is the height or the opposite side and b is the base or the adjacent side. (I've attached a picture just in case) (It might look confusing but the way I've lettered them is taking the 90° angle to be on the right side)
- The a and b doesn't actually matter so don't think too much about which side is a and which one is b, they're more or less the same.
The reason I said that this question might not solely require you to use the Pythagoras theorem is because you need to have two variables for the equation to work, maybe this is what got you confused.
For this question, you're required to use Tangents, Sines and Cosines, or you can combine this method and Pythagoras method.
- Looking for x and y:
Method one: Using Tangents, Sines and Cosines AND the Pythagoras theorem
Step 1: Looking for x(Using Sin, Cos and Tan)
To find the length of side x, we need to use the Cosine of 32°
Cos 32° = Adjacent side ÷ Hypotenuse
0·8480 = x/16
Make x the subject and multiply both sides by 16
0·8480 × 16 = x/16 × 16
13·568 = x
∴ side x is 13·568
Step 2: Looking for y( using the Pythagoras theorem)
a² + b² = c²
a = y, b = x = 13·568, c = 16
y² + 13·568² = 16²
y² + 184·0906 = 256
y² = 256 - 184·0906
y² = 71·9094
y = √71·9094
y = 8·4799
Method 2: Using Tan, Cos and Sin only
Step 1 remains the same as shown above; x = 13·568
Step 2: Looking for y
Y can be gotten by using the Sine of 32°
Sin 32° = Opposite ÷ Hypotenuse
Sin 32° = y ÷ 16
0·5299 = y/16
Make y the subject and divide both sides by 16
0·5299 × 16 = y/16 × 16
8·4784 = y
∴ y is equal to 8·4784
(I know there's a difference in points of the two values of y, but if you round them off to 2 decimal places we see it remains 8·47
2. Looking for z:
Using addition and subtraction
The simpler method would be to just add all the internal angles.
- The little square in the corner tells us that that is a 90° angle
- The internal angles of a triangle ALWAYS add up to 180
32 + 90 + z = 180
122 + z = 180
z = 180 - 122
z = 58
∴ z is equal to 58°
Using Tan, Sin and Cos
The cosine of Z is equal to y divided by 16, where y = 8·47
Cos z = 0·5294
Make z the subject and divide both sides by Cos.
z = 0·5294/Cos
z = 58·04° (I just used the anti-cos to get this number. Press shift, and cos on your calculator then type 0.5294 in front of it)
Here there is also a slight difference in points between the two steps but if you round off step 2 to a whole number you get 58, so it's all good :)
HOPE THIS HELPS!!! :)