you are standing on one side of the river. Figure out the width of the river. All calculations have to be made on the side you are standing on. You cant step into the river or cross it. You may use the 2 laws ( sine and cos ). no guessing allowed. has to be accurate. The calculation that are done should be done by someone who is in grade 10. you cannot use Similar and congruent triangles not allowed

**i understand this question has been answered twice before but the frst time similar triangles were being used which is not allowed and 2nd time to figure out the angle. guessing was used which is again not allowed.

the answer has to be without guess and no use of similar triangles or congruent. 2 laws allowed

You have to measure an angle, and a distance to do this. The last example I gave you is as accurate as you can measure the base distance along the river, and as accurate as one can measure angles with a protractor. Your comment it was just an estimate baffles me, it reflects on someones understanding of measurement and trig.

Good luck.

Just Wondering,

Have you read the whole text?

Item 5 asks for an estimate of the width of the river (not a guess).

Item 7 requires a measurement of the angle using the protractor. In fact, the same measurement is repeated for a check. This is what surveyors do.

The distance is measured with a tape.

This use of the protractor is based on what a surveyor does, who uses a theodolite (measures to within 1-3 seconds of the arc). If you decide that a compass is acceptable, go right ahead. It would give the same order of accuracy as a protractor. Both give measurements, not guesses.

Also, do not forget to mention that the distance between the two points on the base line has to be the horizontal distance, which again is what surveyors do. The distance on the slope will give erroneous results for the width.

To calculate the width of the river without stepping into or crossing it, you can use the laws of sine and cosine along with a few basic trigonometric principles. Here's a step-by-step explanation of how to do it:

1. Stand on one side of the river, and choose a reference point that's directly across from you on the other side. Let's call this point B.

2. Choose another point on your side of the river that's closer to the water's edge, and name it A.

3. Measure the distance between points A and B on the ground. Let's call it AB.

4. Measure the angle between the line AB and the line that's parallel to the riverbank. Let's call it θ (theta). You can use a protractor to measure this angle.

5. From point A, measure the horizontal distance to a point C that lines up with point B. Let's call it AC.

6. Now, we're going to use the law of sine to find the width of the river. The law of sine states that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides of the triangle.

7. In our case, we have a right triangle ABC, where the angle θ is opposite side AB, and the side AC is adjacent to angle θ. Since we can't step into the river, we can't directly measure the length of side BC, which is the width of the river. However, we can find it indirectly using the law of sine, as stated in the previous step.

8. According to the law of sine, we have the following ratio: AB/sin(θ) = BC/sin(90° - θ).

9. Since sin(90° - θ) = cos(θ), we can simplify the equation: AB/sin(θ) = BC/cos(θ).

10. Rearrange the equation to solve for BC: BC = AB * cos(θ) / sin(θ).

11. Substitute the values you measured into the equation: BC = AB * cos(θ) / sin(θ).

12. Calculate BC using a calculator with trigonometric functions: BC = AB * cos(θ) / sin(θ).

By following these steps, you can accurately calculate the width of the river using the laws of sine and cosine, without relying on similar or congruent triangles or making any guesses.