Respuesta :
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
Unit 8: Homework 2; Special Right Triangles
Correct responses;
- x = 13, y = 13·√2
- x = 15·√2, y = 15·√2
- x = 6, y = 3·√3
- x = 17·√3, y = 17
- x = 10, y = 10
- x = 50, y = 25
- x = 2·√17, y = 2·√17
- x = 16·√3, y = 8·√3
- x = 33, y = 11·√3
- x = 3·√2, y = 2·√6
- x = √10, y = 2·√5
- x = 4·√7, y = 8·√7
- x = 17·√3, y = 34·√2, z = 34
- x = 18·√3, y = 18, z = 9
Methods used for finding the above values
Solutions:
1. An acute angle of the right triangle is 45°, therefore, the triangle is an isosceles triangle, and the leg lengths are equal.
Therefore;
- x = 13
The length of the hypotenuse side of an isosceles right triangle = A leg length × √2
Therefore;
- The length of the hypotenuse side, y = 13·√2
2. An interior angle of the triangle = 45°
Therefore;
x = y
30 = x·√2
Which gives;
[tex]x = \dfrac{30}{\sqrt{2} } = \dfrac{30 \cdot \sqrt{2} }{2} = \mathbf{15 \cdot \sqrt{2}} = y[/tex]
- x = y = 15·√2
3. The interior angle adjacent to the leg of length 3 = 60°
Therefore;
The hypotenuse side, x = 2 × adjacent leg length
Which gives;
- x = 2 × 3 = 6
In a right triangle having an interior angle of 60°, we have;
2 × Opposite leg length = √3 × The length of the hypotenuse
Therefore;
2 × y = √3 × x
Which gives;
2 × y = √3 × 6
- y = 3·√3
4. The leg lengths are, x and y
An interior angle opposite to the leg length y is 30°
The hypotenuse side = 34
The length of the hypotenuse side = 2 × The leg length opposite the 30° angle
Therefore;
34 = 2 × y
[tex]y = \dfrac{34}{2} = \mathbf{17}[/tex]
- y = 17
The leg with length x is adjacent to the 30° angle, which gives;
2·x = √3 × 34
- x = 17·√3
5. An acute interior angle is 45°
Therefore, the relationship are;
x = y
x·√2 = 10·√2
Which gives;
- x = 10 = y
6. An acute interior angle is 60°, which in relation to the position of the sides gives;
x·√3 = 2 × 25·√3
Therefore;
x = 2 × 25 = 50
x = 50
y = x ÷ 2
Therefore;
y = 50 ÷ 2 = 25
- y = 25
7. An acute interior angle is 45°, which gives;
x·√2 = 2·√14
[tex]x = \dfrac{2 \cdot \sqrt{14} }{\sqrt{2} } = \sqrt{2} \times \sqrt{14} = \sqrt{28} = 2 \cdot \sqrt{7} [/tex]
- x = 2·√7
- y = 2·√7
8. An interior angle is 30°
With regards to the location of the variables, we have;
2 × 24 = x·√3
Therefore;
- x = 16·√3
2·y = x
Therefore;
[tex]y = \dfrac{x}{2} [/tex]
Which gives;
[tex]y = \dfrac{16 \cdot \sqrt{2} }{2} = \mathbf{8 \cdot \sqrt{2} }[/tex]
- y = 8·√2
9. An interior angle is 60°, which gives;
[tex]y = \dfrac{22 \cdot \sqrt{3} }{2} = 11 \cdot \sqrt{3} [/tex]
- y = 11·√3
√3 × 22·√3 = 2 × x
x = 11·√3 × √3 = 33
- x = 33
10. An angle of the right triangle is 30°
With respect to the location of the variables, we have;
y = 2 × √6 = 2·√6
- y = 2·√6
y·√3 = 2 × x
Therefore;
2·√6 × √3 = 2 × x
x = √(18) = 3·√2
- x = 3·√2
11. Right triangle with a 45° interior angle
- x = [tex]\underline{\sqrt{10} }[/tex]
y = √(10) × √2 = √(20) = 2·√5
- y = 2·√5
12. Interior angle of the right triangle is 60°
y·√3 = 2 × 4·√(21) = 8 × √7 × √3
- y = 8·√7
[tex]x = \dfrac{y}{2} [/tex]
Therefore;
[tex]x = \dfrac{8 \cdot \sqrt{7} }{2} = \mathbf{ 4 \cdot \sqrt{7} }[/tex]
- x = 4·√7
13. The ratio of the adjacent leg to the opposite leg to the 30° angle is √3, therefore;
- x = 17·√3
The hypotenuse side = 2 × 17 = 34
The hypotenuse side of the right triangle having a 30° angle is a leg in
the right triangle that has the 45° angle, therefore;
- z = 34
- y = 34·√2
14. In the 60° right triangle, we have;
x·√3 = 2 × 27 = 54 = 18 × 3
- x = 18·√3
[tex]Length \ of \ the \ common \ side = \dfrac{x}{2} [/tex]
Which gives;
[tex]Length \ of \ the \ common \ side = \dfrac{18 \cdot \sqrt{3} }{2} = 9 \cdot \sqrt{3} [/tex]
Length of the common side = 9·√3
In the 30° right triangle, we have;
x·√3 = 9·√3
Therefore;
- z = 9
y = 2·x
Therefore;
- y = 2 × 9 = 18
Learn more about special right triangles here:
https://brainly.com/question/10635635
