Need help with special right triangles
1 answer:
Step-by-step explanation:
For solving this, it's important to know basic trigonometric functions:
sin = opposite side / hypotenuse
cos = adjacent side / hypotenuse
tan = opposite side / adjacent side
It's also important to know that it is necessary to know their values for special right triangles and angles of 30°, 45° and 60°
1. We are given angle if 45° and its adjacent side of 13. We want to know the opposite side (x) and hypotenuse (y). So:
tan = opposite side / adjacent side
tan 45 = x / 13
1 = x / 13
x = 13
Remember that tangent of 45° angle is 1.
We found the opposite side to be also 13.
To find y we can use:
sin = opposite side / hypotenuse
sin 45 = 13 / y
(√2)/2 = 13 / y
y = 13√2
Important to remember, sin 45 and cos 45 are (√2)/2
2. We are given 45° angle and hypotenuse of 30. We need to find opposite side (x) and adjacent side (y).
sin = opposite side / hypotenuse
sin 45 = x / 30
(√2)/2 = x / 30
x = 15√2
Also:
cos = adjacent side / hypotenuse
cos 45 = y / 30
(√2)/2 = y / 30
y = 15√2
We can conclude that in the right triangle, when angle is 45°, opposite and adjacent sides are equal.
3. We are given angle of 60° and adjacent side of 3. We need to find opposite side (y) and hypotenuse (x).
cos = adjacent side / hypotenuse
cos 60° = 3 / x
1/2 = 3 / x
x = 6
Note that cos 60° equals 1/2.
sin = opposite side / hypotenuse
sin 60° = y / 6
(√3)/2 = y / 6
y = 3√3
Note that sin 60° equals (√3)/2.
4. We are given angle of 30° and hypotenuse of 34. We need to find opposite side (y) and adjacent side (x).
sin = opposite side / hypotenuse
sin 30° = y / 34
1/2 = y / 34
y = 17
Note that sin 30° equals to cos 60° equals 1/2.
cos = adjacent side / hypotenuse
cos 30° = x / 34
(√3)/2 = x / 34
x = 17√3
Remember that cos 30° equals to sin 60° which is (√3)/2.
5. We are given an angle of 45° and hypotenuse of 10√2. We need to find the opposite side (y) and adjacent side (x).
sin = opposite side / hypotenuse
sin 45° = y / 10√2
(√2)/2 = y / 10√2
y = 10
To save time, we already said that opposite and adjacent sides are equal in right triangle with 45° angle, so x = y = 10.
6. We are given angle of 60° and opposite side of 25√3. We need to find adjacent side (y) and hypotenuse (x).
sin = opposite side / hypotenuse
sin 60° = 25√3 / x
(√3)/2 = 25√3 / x
x = 50.
tan = opposite side / adjacent side
tan 60° = 25√3 / y
√3 = 25√3 / y
y = 25.
Remember that tangent of 60° angle is √3.
7. We are given angle of 45° and hypotenuse of 2√14. We need to find adjacent side (x) and opposite side (y).
cos = adjacent side / hypotenuse
cos 45° = x / 2√14
(√2)/2 = x / 2√14
x = √28
Again, adjacent and opposite sides are equal in 45° right triangle, so x = y = √28.
8. We are given an angle of 30° and an adjacent side of 24. We need to find the opposite side (y) and hypotenuse (x).
tan = opposite side / adjacent side
tan 30° = y / 24
(√3)/3 = y / 24
y = 8√3
sin = opposite side / hypotenuse
sin 30° = 8√3 / x
1/2 = 8√3 / x
x = 16√3
Note that tan = sin/cos, so tan 30° = sin 30° / cos 30°
tan 30° = 1/2 / (√3)/2 = (√3)/3
I'm showing you this, so you don't have to memorize tan for special angles, you can find it from sin and cos.
9. We are given an angle of 60° and hypotenuse of 22√3. We need to find the opposite side (y) and adjacent side (x).
sin = opposite side / hypotenuse
sin 60° = y / 22√3
(√3)/2 = y / 22√3
y = 33
cos = adjacent side / hypotenuse
cos 60° = x / 22√3
1/2 = x / 22√3
x = 11√3
10. We are given an angle of 30° and the opposite side of √6. We need to find the adjacent side (x) and hypotenuse (y).
sin = opposite side / hypotenuse
sin 30° = √6 / y
1/2 = √6 / y
y = 2√6
tan = opposite side / adjacent side
tan 30° = √6 / x
(√3)/3 = √6 / x
x = √18
11. We are given an angle of 45° and an adjacent side of √10. We need to find the opposite side (x) and hypotenuse (y).
In this triangle opposite and adjacent sides are equal (45° angle), so x = √10
sin = opposite side / hypotenuse
sin 45° = √10 / y
(√2)/2 = √10 / y
y = √20 = 2√5
12. We are given an angle of 60° and the opposite side of 4√21. We need to find the adjacent side (x) and hypotenuse (y).
sin = opposite side / hypotenuse
sin 60° = 4√21 / y
(√3)/2 = 4√21 / y
y = 8√7
tan = opposite side / adjacent side
tan 60° = 4√21 / x
√3 = 4√21 / x
x = 4√7
13. Now things get a little trickier. We are given an angle of 30° and the opposite side of 17. We need to find the adjacent side (x).
Note that, at the moment, we are only dealing with this smaller triangle.
tan = opposite side / adjacent side
tan 30° = 17 / x
(√3)/3 = 17 / x
x = 17√3
Now, note that hypotenuse of the smaller triangle is the same length as the side z (adjacent and opposite side of 45° angle)
With Pythagorean theorem, we can find the hypotenuse of the smaller triangle, which equals to z.
z = √(17^2 + (17√3)^2)
z = 34
And now, finally to find y (hypotenuse of bigger triangle). We are given 45° angle and adjacent side z (34).
cos = adjacent side / hypotenuse
cos 45° = 34 / y
(√2)/2 = 34 / y
y = 34√2
14. Photo added
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