# Cos/Sin Rule Exercises: Challenges and Solutions

• Rukawa0320
In summary, the conversation discusses difficulties with exercises involving the cos/sin rule and solving problems involving quadrilaterals and angles. The radius of a circumscribing circle and the length of a side are sought in a cyclic quadrilateral, and the height of a tower is determined using angles and distances on a slope. The speaker has attempted the problems but is seeking help in finding a solution.
Rukawa0320
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

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?

## 1. What are the cos/sin rules and why are they important in scientific calculations?

The cosine and sine rules are mathematical formulas used to calculate the lengths of sides and angles in a triangle. They are important in scientific calculations because triangles are a fundamental shape in geometry and are used to model many real-world situations, such as forces, vectors, and waves.

## 2. How do I know when to use the cos rule versus the sin rule?

The cosine rule is used when you know two sides and the included angle of a triangle, and need to find the third side. The sine rule is used when you know two angles and a side, and need to find another side or angle. Remember the mnemonic SOH-CAH-TOA: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse.

## 3. What are some common challenges when applying the cos/sin rules?

One of the most common challenges is remembering which formula to use in a particular situation. Another challenge is understanding how to solve for a missing side or angle when the given information is not in the same units (e.g. degrees vs. radians). Additionally, rounding errors can occur when using calculators, so it's important to round to the appropriate number of significant figures.

## 4. Can you provide an example of a cos/sin rule exercise and its solution?

Example: Find the length of side c in the triangle below, given that angle A = 60°, side b = 10, and side a = 8.Solution: Using the sine rule, we can set up the following equation: sin(A)/a = sin(B)/b. Plugging in the known values, we get sin(60°)/8 = sin(B)/10. Solving for sin(B), we get sin(B) = (10/8)sin(60°) = 1.299. Taking the inverse sine, we get B = 51.5°. Finally, we can use the cosine rule to find c: c^2 = a^2 + b^2 - 2abcos(C), where C is the remaining angle. Plugging in the known values, we get c^2 = 8^2 + 10^2 - 2(8)(10)cos(68.5°) = 164.9. Taking the square root, we get c ≈ 12.84.

## 5. How can I check my answers when working on cos/sin rule exercises?

One way to check your answers is to use a calculator or trigonometry table to find the sine, cosine, and tangent of the given angles and sides. Make sure your answers are consistent with these values. Another way is to use the Pythagorean theorem to check if your calculated values for the sides of a triangle satisfy the equation a^2 + b^2 = c^2.

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