8. The length of the support is 7. 9 m
9. The length of the conveyer is 12m
3. a = 59m
A = 44°
B = 52°
4. c = 88. 6 mm
b = 49. 1 m
B = 34°
<h3>How to solve the trigonometry</h3>
8. We have the angle to be 20 degrees
Opposite side = x
hypotenuse = 23m
Using the sine ratio
sin θ = opposite/ hypotenuse
![sin 20 = \frac{x}{23}](https://tex.z-dn.net/?f=sin%2020%20%3D%20%5Cfrac%7Bx%7D%7B23%7D)
Cross multiply
× ![23 = x](https://tex.z-dn.net/?f=23%20%3D%20x)
× ![23](https://tex.z-dn.net/?f=23)
x = 7. 9 m
The length of the support is 7. 9 m
9. The angle of elevation is 37. 3 degrees
Hypotenuse = 19 . 0m
Opposite = x
Using the sine ratio
sin θ = opposite/ hypotenuse
![sin 37. 3 = \frac{x}{19}](https://tex.z-dn.net/?f=sin%2037.%203%20%3D%20%5Cfrac%7Bx%7D%7B19%7D)
cross multiply
× ![19](https://tex.z-dn.net/?f=19)
x = 11.5
x = 12 m in 2 significant figures
The length of the conveyer is 12m
3. To determine the sides and angles, we use the sine rule;
![\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7Bsin%20A%7D%20%3D%20%5Cfrac%7Bb%7D%7Bsin%20B%7D%20%3D%20%5Cfrac%7Bc%7D%7Bsin%20C%7D)
For side a, we use the Pythagorean theorem
![c^2 = a^2 + b^2](https://tex.z-dn.net/?f=c%5E2%20%3D%20a%5E2%20%2B%20b%5E2)
![85^2 = a^2 + 67^2](https://tex.z-dn.net/?f=85%5E2%20%3D%20a%5E2%20%2B%2067%5E2)
![a = \sqrt{89^2-67^2}](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%7B89%5E2-67%5E2%7D)
![a = \sqrt{3432}](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%7B3432%7D)
a = 58. 58, a = 59m
To find angle A and B, use the sine rule
![\frac{59}{sin A } = \frac{85}{sin 90}](https://tex.z-dn.net/?f=%5Cfrac%7B59%7D%7Bsin%20A%20%7D%20%3D%20%5Cfrac%7B85%7D%7Bsin%2090%7D)
cross multiply
×
=
× ![59](https://tex.z-dn.net/?f=59)
make sin A subject of formula
![sin A = \frac{59}{85}](https://tex.z-dn.net/?f=sin%20A%20%3D%20%5Cfrac%7B59%7D%7B85%7D)
![sin A = 0. 6941](https://tex.z-dn.net/?f=sin%20A%20%3D%200.%206941)
A = ![sin^-^1(0. 6941)](https://tex.z-dn.net/?f=sin%5E-%5E1%280.%206941%29)
A = 44°
![\frac{67}{sin B} = \frac{85}{sin 90}](https://tex.z-dn.net/?f=%5Cfrac%7B67%7D%7Bsin%20B%7D%20%3D%20%5Cfrac%7B85%7D%7Bsin%2090%7D)
cross multiply
×
=
× ![67](https://tex.z-dn.net/?f=67)
make sin b subject of formula
![sin B = \frac{67}{85}](https://tex.z-dn.net/?f=sin%20B%20%3D%20%5Cfrac%7B67%7D%7B85%7D)
![sin B = 0. 7882](https://tex.z-dn.net/?f=sin%20B%20%3D%200.%207882)
B = ![sin^-^1( 0. 7882)](https://tex.z-dn.net/?f=sin%5E-%5E1%28%200.%207882%29)
B = 52°
4. To find the sides, we use the sine rule;
![\frac{74. 0}{sin 56. 6} = \frac{c}{sin 90}](https://tex.z-dn.net/?f=%5Cfrac%7B74.%200%7D%7Bsin%2056.%206%7D%20%3D%20%5Cfrac%7Bc%7D%7Bsin%2090%7D)
Cross multiply
×
× ![74](https://tex.z-dn.net/?f=74)
make 'c' subject of formula
![c = \frac{74}{0. 8348}](https://tex.z-dn.net/?f=c%20%3D%20%5Cfrac%7B74%7D%7B0.%208348%7D)
c = 88. 6 mm
To find length b, we use the Pythagorean theorem
![c^2 = a^2 + b^2](https://tex.z-dn.net/?f=c%5E2%20%3D%20a%5E2%20%2B%20b%5E2)
![b^2 = c^2 - a^2](https://tex.z-dn.net/?f=b%5E2%20%3D%20c%5E2%20-%20a%5E2)
![b^2 = 88. 8^2 - 74^2](https://tex.z-dn.net/?f=b%5E2%20%3D%2088.%208%5E2%20-%2074%5E2)
![b = \sqrt{7885. 44 - 5476}\\\\ b = \sqrt{2409. 44}](https://tex.z-dn.net/?f=b%20%3D%20%5Csqrt%7B7885.%2044%20-%205476%7D%5C%5C%5C%5C%20b%20%3D%20%5Csqrt%7B2409.%2044%7D)
b = 49. 1 m
![\frac{74. 0}{sin 56. 6} = \frac{49. 1}{sin B}](https://tex.z-dn.net/?f=%5Cfrac%7B74.%200%7D%7Bsin%2056.%206%7D%20%3D%20%5Cfrac%7B49.%201%7D%7Bsin%20B%7D)
cross multiply
![sin B = \frac{40. 99}{74. 0}](https://tex.z-dn.net/?f=sin%20B%20%3D%20%5Cfrac%7B40.%2099%7D%7B74.%200%7D)
B = ![sin^-^1(0. 5539)](https://tex.z-dn.net/?f=sin%5E-%5E1%280.%205539%29)
B = 34°
Learn more about trigonometric identity here:
brainly.com/question/7331447
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