trigonometric functions application..problem solving?

Question

1. A 36-ft ladder rests against the edge of a roof. Four feet of the ladder's length extend beyond the edge of the roof, which is 28 ft above the ground level. Find the tangent and secant of the angle that the makes with the ground.

2. The angle of elevation from a point on the ground to the top of the pyramid is 35degrees 30 minutes. The angle of elevation from a point 135ft farther back to the top of the pyramid is 21 degrees. Find the height of the pyramid.

Answer 1

If 4 ft of ladder beyond edge of roof, the hypotenuse is 32 ft. Height=28 ft.
sin theta=o/h
theta=sin^-1(28/32)
=61.04°

tangent of the angle:
tan 61.04°=1.81

secant of angle
sec 61.04°
=1/cos(61.04°)
=2.07

2. generate 2 equations. let a=length of first point to puramid
tan theta=o/a
tan 35.5°=o/a
a=o/(tan 35.5°)

tan 21°=o/(a+135)
a+135=o/(tan 21°)
a=o/(tan 21°)-135

equate them
o/(tan 35.5)=o/(tan 21)-135
o/0.7133=o/0.3839-135
1.402o=2.605o-135
1.203o=135
o=112.2 ft
so height=112.2 ft

Answer 2

1. First identify the sides of the "triangle" in the figure

a = 28 ft (height of the roof above the ground)
b = horizontal distance from the ladder to the base of the house
c = hypotenuse or the length of ladder (36 - 4 = 32ft)

Using Pythagorean Theorem a^2 + b^2 = c^2
b = sqrt (c^2 - a^2)
b = 15.5 ft

Let x be the angle of elevation the ladder makes with the ground

Tan x = opp side / adj side or a/b
Tan x = 1.80645
x = arctan a/b = arctan 28/15.5
x = 61 deg

Sec x = Hyp / adj side or c/b
Sec x = 2.0645
of course, x = arcsec c / b = 61 deg

2. This can be solved using figures and principles of sine and cosine laws or using 2 equations with 2 unknowns. Difficult to write down the process but the answer is 112.21 ft

Answer 3

1. The bottom 32ft of the ladder forms the hypotenuse of the triangle.
28/32 is the sin of the angle that the ladder makes with the ground.
tanX = tan(arcsin(28/32)) = 1.80739
secX = sec(arcsin(28/32)) = 2.06559

2. Label the top of the pyramid A, call the vertex of the 21 degree angle B, and the vertex of the 35.5 degree angle C. These three points form a triangle. Angle C = 180 - 35.5 = 144.5 and use sine law:
AB/sinC = BC/sinA
AB = 135sin144.5/sin14.5 = 313.104m
AB is also the hypotenuse of the right triangle that has the height of the pyramid as one leg.
313.104tan21 = 120.189m

Answer 4

hint: enable t = cos(x) the equation turns into t^2 +t =0 t(t +a million) =0 t =0 or t =-a million remedy cos(x) =-a million ----> x = (2n+a million)Pi (n =0, +/-a million; +/-2;......) cos(x) =0 x = pi/2 + 2kpi (ok =0, +/-a million; +/-2 ;.....)