Before Starting the Problem, Remember in Any Triangle Sum of Angles Are 180Deg

Basic rules

Before starting the problem, remember in any triangle sum of angles are 180deg

A+C+B=180

A’+B’+C=90

A’=A, B’=B

To obtain a good results Please work these problems in a group. Do not look at the solution, after you spend at least half an hour on each problem look at the solution.

Problem 1:

In the triangle ABC, the line AB is along a straight riverbank. We measure the distance AB as 118 meters, and angles A and B are 63° and 55° . What is the distance b = AC?

Do not look at the solution:

Before solving this problem study the Law of Sines in your lecture very carefully.

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Problem 2

You are traveling uphill on a road and see a sign telling you this is a 5% grade, i.e. rising 5 meters for every 100 meters of road. What is the angle between the road and the horizontal direction.

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Before solving This problem Study the vectors and Trigonometry.

Problem 3

An airplane is flying at 170 km/s towards the north-east, in a direction making an angle of 52° with the eastward direction.

The wind is blowing at 30 km/s towards the north west, making an angle 20° with the northward direction. What is the actual "ground speed" of the airplane, and whatis the angle A between the airplane's actual path and the eastward direction?

Problem No 4 (Study the law of sines)

In problem no 1 what is the perpendicular distance from C to the line c = AB

Problem No 5 (Back ground)

When a beam of light hits the surface of a flat piece of glass, it is generally bent by some angle. Draw a line perpendicular to the point on the surface where the beam enters. Then if the beam reaches the surface along a path making an angle A with the surface, it continues inside the glass with an angle making an angle B, where

sin B = (sin A)/n

The number n ("refractive index") is a property of the glass and is larger than one. See the figure below;

Problem No 5

Given values of A=0, 20, 40, 60, and 80 degrees, and n = 1.45, what is B in each case?

Note

If this formula somehow fails, the beam cannot leave the glass but is reflected from the boundary surface back into the glass, like from a mirror ("total internal reflection")

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Problem No 6

If cos X = 2 sinX, what is sin X? What is the angle X?

Problem no 7

Find the hight of the tree if AB=45m and X=65degree

Problem no 8

If x=34.87deg (angle of depression) , how fr is the ship from the land. Assume the hight of tower is 200m.

Solution to problem 1

Because the sum of all angles is 180 degrees, the angle C must equal 62°. Then by the law of sines

118/sin(62°) = b/sin(55°)

Multiply both sides by sin(55) to obtain the length b = AC.

Solution to Problem 3

Let us denote the velocity vector of the airplane relative to the air as V, that of the wind relative to the ground as W, and the velocity of the airplane relative to the ground U=V+W, where the addition is one of vectors. Draw a diagram with the given speeds and angles labeled appropriately.

To perform the actual addition each vector must be resolved into its components. We get

Vx = 170 cos(52°) = 104.6 Vy = 170 sin(52°) = 133.96

Wx = -30 sin(20°) = -10.26 Wy = 30 cos(20°) = 28.19

Add:

Ux = 94.4 Uy = 162.15

From Pythagoras, since U2 = Ux2 + Uy2, U= 187.63 km/hr

Therefore

cos A = Ux /U = 0.503125

Using the cos–1 button

A = 59.8°