## Pages

### Group 2

Group members: Jia Qi, Howe Wee, Dean, Chang Han, Carissa

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Problem:

Two men facing a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be h =120 m, find the distance (in meters) between the two men.

Assumptions:

- The men and the tall building all lie on the same plane
- The men are of the same height and are looking up from the same height
- The men are at the same elevation
- Two men are standing on opposide sides of the building OR they are standing at the same sides of the building OR the two men are standing perpendicular to each other

Solutions:

1) Assuming that the two men are on the opposite sides of the building.

2) Assuming that the two men are standing at the same sides of the building.

3) Assuming that the two men are standing perpendicular to each other.

tan 30 = 120/a
a = 120/tan30
a = 207.8 (4sf)

tan 60 = 120/a
a = 120/tan60
a = 69.28 (4sf)

69.28^2 + 207.8^2 = x^2
x = sqrt (69.28^2 + 207.8^2)
x = 219 (3sf)

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Sine Rule:

Initially, A and a do not have an obvious relationship, so it is difficult to find the equation that connects them. Sin rule makes use of the common line (h) to find the equation between sinB and sinA. Thus bsinA = h and asinB = h, sinA/a = sinB/b. To find the relationship between sinC and sinB and sinA another line is drawn from B to line AC which is used as the common value which can be used in the same way to find the ratio between sinC and sinA. Thus using the same method, sinA/a = sinB/b = sinC/c.

Cosine Rule:

Get a triangle. Perpendicularly bisect it from one point to another segment. Name them as shown in the picture. AB = c, BC = a, AC = b, C to the point where the bisector intersects the segment as h, that point to A as x, and B to that point as c-x. (Refer to diagram for clearer visualisation)
Use pythagorus theorem to form two different equations of the two right angle triangles, B-C-intersection, and C-A-intersection.
b^2 = h^2 + x^2 and
a^2 = h^2 + (c-x)^2
Expand the 2nd equation and make x^2 + h^2 the subject. From there, equate b^2 to the expanded form, a^2-c^2+2cx.
Use trigonometry to find the equation cosA=x/b , bcosA = x
Substitute x into the equation to get a^2 = b^2 + c^2 - 2bc(cosA).-

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Bearing Question

A = 60˚ + 50˚ = 110˚

a^2 = b^2 + c^2 - 2bc cos(A)
a^2 = 3^2 + 5^2 - 2 x 3 x 5 cos(110˚)

a^2 = 9+25 - 30 cos 110˚
a = 6.65 (3sf)

6.65/ sin110˚ = 3/sin B
6.65 sin B = 3 sin110˚
(3 sin110˚ )/6.65 = sin B
B = sin^-1 ((3 sin110˚ )/6.65)
B = 25.1 (1dp)

90˚ - 25.1˚ - 50˚ = 14.9˚

Ans: 14.9˚T for 6.65km.

### GRAPH SKETCHINGANALYSIS OF SINE GRAPHS

Activity 1:

Define the following terms, where possible identify the context when it is applied.
A. Amplitude

The maximum displacement of the particles from the rest position.

B. Wavelength

The shortest distance between any 2 points in a wave that are in phase.

C. Frequency

Number of complete wave per second.

D. Phase

Any two points that move in the same direction and have the same speed and the same displacement from the rest position.

E. Characteristics of the Wave

- It transfers energy away from the wave source.
- Wave transfers away energy from one point to another but it does not transfer matter.
- It is made up of periodic motion which is repeated at regular interval.

Activity 2:
Study the following graphs with reference to y = sinx and (1) identify the transformation (s) of the graphs eg. enlargement, shift in x-direction. (2) State whether the new graph is similar to y = sinx. (3) suggest possible the 'transformed' equation. Note that x is in radian.
Transformation 1 1. Enlargement in the y axis direction (Amplitude increase) 2. It has a larger amplitude, but still the same frequency (period). 3. y = 3sinx
Transformation 2 1. X - intercept frequency increased 2. It has an increased frequency, but still the same amplitude 3. y = sin(2x)
Transformation 3 1. Shifted in x-axis direction 2. It has 'moved' to the right, but has the same frequency and amplitude 3. y = sin(x-1)
Transformation 4 1. Shifted in the y-axis direction 2. It has moved up, but has the same frequency and amplitude 3. y = sinx + 3