5.6 Right Triangle Trigonometry

The angle of elevation of the top of a water tower from point A on the ground is 19.9°. From point B, 50.0 feet closer to the tower, the angle of elevation is 21.8°. What is the height of the tower?

At point B, tan21.8° = y/x or x= y/tan21.8° Since the distance to the base of the tower from Point A is x+50 or y=(x+50)tan19.9° To find the value of y we must write an equation that involves only y. Since x = y/tan21.8°, we can substitute y/tan21.8° for x in the last equation. y= (y/tan21.8° + 50) tan19.9° y= y×tan19.9°/21.8° +50tan19.9° y-y × tan19.9/tan21.8 = 50 tan19.9° y(1- tan19.9°/tan21.8° = 50 tan19.9° y= 50 tan 19.9°/1- tan19.9°/tan21.8°≈ 191 feet

A guy wire of length 108 meters runs from the top of an antenna to the ground. If the angle of elevation of the top of the antenna, sighting along the guy wire, is 42.3°, then what is the height of the antenna?

Let y represent the height of the antenna, since sin(42.3°) = y/108 y = 108 × sin(42.3°) ≈ 72.7 meters

Powers was photographing the Soviet Union from U-2 at an altitude of 14 miles. How wide a path on the earth's surface could Powers see from that altitude? (Use 3950 miles as the earth's radius)

Since a line tangent to a circle (line of sight) is perpendicular to the radius at the point of tangency, the angle α at the center of the earth is an acute angle of a right triangle with hypotenuse 3950 + 14 or 3964. So we have cosα= 3950/3964 α=cos-¹(3950/3964) ≈ 4.8° The width of the path seen by Powers is the length of the arc intercepted by the central angle 2α or 9.6°. Using the formula s=αr where α is in radians. s=9.6° deg × π rad/180 deg. × 3950 miles = 661.8 miles

Solve the right triangle in which a=4 and b=6. Find the acute angles to the nearest tenth degree.

r²= a²+b² r= 2√13 sin= 4/2√3 = 2√13 Now use the inverse of sin to find the alpha degree sin-¹(2√13) = 33.7° α + β = 90 33.7 + β = 90 β= 56.3° The angles of the triangle are 33.7°,56.3°, and 90°, and the sides opposite those angles are 4, 6, and 2√13.

Find the values of all six trigonometric functions for the angle α of the right triangle with legs of length 1 and 4. 1 being the opposite and 4 being the adjacent.

sin= √17/17 cos= 4√17/17 tan= 1/4 csc= √17 sec= √17/4 cot= 4

Find the values of the six trigonometric functions of the angle α in standard position whose terminal side passes though (4,-2).

sinα = -√5/5 cosα= 2√5/5 tanα= -2 cscα= -5/√5 secα= √5/2 cotα= -2

theorem trig ratio for cosα

x/r

theorem trig ratio for sinα

y/r

theorem trig ratio for tanα

y/x (x≠0)

theorem ratio of a right angle of tanα

opp/adj

theorem ratio of a right angle of sinα

opp/hyp

definition of angle of elevation

point above the horizontal line

definition of angle of depression

point below the horizontal line

theorem ratio of a right angle of cosα

adj/hyp