Cos/Sin Rule Exercises: Challenges and Solutions

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The discussion focuses on challenges faced while solving exercises involving the cosine and sine rules, particularly in a cyclic quadrilateral and elevation problems. The user successfully calculated the length of side AD in a cyclic quadrilateral but struggled with applying Thales's theorem and finding the radius of the circumscribing circle. For the second problem, they determined the time for a barge to reach a bridge using angle measurements, estimating it to be around 19.2 minutes. The third problem involved calculating the height of a tower on a slope, where the user reached a total height of 25.37 meters but could not progress further. The conversation highlights the importance of visual aids and understanding trigonometric relationships in solving these types of problems.
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Hey guys,

We've been learning cos/sin rule since the 1st of school, and i really had difficulties with the following exercises, I've been working them yesterday for 2-3 hours and could only manage to do part of it...

1. IN Quadrilateral ABCD, AB=7 cm, BC= 8cm, CD=5cm and angle ABC=52. Given that ABCD is a cyclic quadrilateral, find the radius of its circumscribing circle and the length of AD.

I was able to calculate the AD, which is about 2,26cm and i also got that the diameters of the quadrilateral are 8cm and 6,63cm. I tried to apply Thales's theorem in the circle, but couldn't really find a solution.

2. From a barge moving with constant speed along a straight canal the angle of elevation of a bridge is 10 degrees. After 10 minutes the angle is 15 degrees. How much longer will it be b4 the barge reaches the bridge, to the nearest second.

3. A tower stand on a slope which is inclined at an angle of 17,2 degrees to the horizontal. From a point further up the slope and 150m from the base of the tower the angle of depresssion of the top of the tower is found to be 9.6 degrees. Find the height of the tower.

I started doing 2 and 3, but couldn't get the idea how to solve it properly, of course i got some extra datas but coundt use it to solve the problem.

Any help appreciated
 
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For #2) What have you determined from the problem? Since the boat is moving at a constant velocity, you can first define as constants the velocity of the boat, and it's position from the bridge at the point where it is 10 degrees from the horizon. Think to yourself: How was 10 degrees determined? 10 degrees = arctan(heightofbridge / distancefrombridge).

3) It will be really helpful to draw a picture of this. Have you constructed what you think is an accurate picture of the situation?
 
for 3) of course i drew a picture, i could only get as far as the slope+tower all together is 25.37m, maybe i misdrew the picture or just couldn't continue from the answer (25.37) i got

and 2) i used your hint and calculated out the height/distance ration, i tried to substitue it into the sin rule but i got 1 in both situation (tan 10 and tan15)
 
i got the answer for 2) (its about 19.2min), does anyone have some hint for the 1st one?
 

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