Aim: How do we apply the trig ratios for indirect measurements?
Objectives: to apply the trig ratios to calculate unknown length when given a right triangle
Do Now: Pick two complementary angle measures (A and B) and evaluate sinA and cos B. What do
you notice about sinA and cos B?
Property: If A + B = 90, then sin A = cos B.
This lesson will focus on word problems that require the use of trig ratios.
EX1: A tree casts a shadow that is 20 feet long. The angle of elevation from the end of the shadow to
the top of the tree is 66°. Determine the distance between the end of the shadow to the top of the tree,
to the nearest foot.
20
x
x cos 66  20
x  49
cos 66 
EX2: As shown in the accompanying diagram, a ladder is leaning against a vertical wall, making an
angle of 70° with the ground and reaching a height of 10.39 feet on the wall.
Find, to the nearest foot, the length of the ladder.
Find, to the nearest foot, the distance from the base of the ladder to the wall.
10.39
ladder
ladder  11 ft
sin(70) 
10.39
base
base  4 ft
tan(70) 
Since the lines of sight are parallel, we conclude
the angle of elevation is congruent to angle of
depression of alternate interior angles theorem.
EX3: A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying
diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the
cliff is 18° and the angle of elevation to the top of the lighthouse is 28°. What is the height of the
lighthouse, x, to the nearest tenth of a foot?
The triangle containing the lighthouse is not a right triangle. We will learn later how to use trig ratios
to solve non-right triangles. At this point, we are only dealing with right triangles.
tan 28 
x  cliff
. In order to solve for x, we need to find out the height of the cliff.
200
cliff
200
cliff  64.98393925
tan18 
tan 28 
x  64.98393925
200
x  41.4
EX4: You were flying a kite on a bluff, but you managed somehow to dump your kite into the lake
below. You know that you've given out 325 feet of string. A surveyor tells you that the angle of
declination from your position to the kite is 15°. How high is the bluff where you and the surveyor are
standing?
I have "opposite", hypotenuse, and an angle, so I'll use the sine ratio to find the height.
h/325 = sin(15°) Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved
h = 325×sin(15°) = 84.11618966...
The bluff is about 84 feet above the lake.
HW#8: P310: 29, 30
P319 – 320: 3, 5, 7, 9
Solutions
P310:
29) WH = 53; TH = 96 TB = BH + TH = 136 ft
30) GF = 222; EF = 230
P319 – 320:
3) h = 50 m
5) h = 2.3 km
7) Martha =65.6
9) x = 440 m
Heidi=128.5
Heidi – Martha = 63 cm
Aim: How do we apply the trig ratios for indirect measurements?
Do Now: Pick two complementary angle measures (A and B) and evaluate sinA and cos B.
Relationship: If A + B = 90, then __________________________.
EX1: A tree casts a shadow that is 20 feet long. The angle of elevation from the end of the shadow to
the top of the tree is 66°. Determine the distance between the end of the shadow to the top of the tree,
to the nearest foot.
EX2: As shown in the accompanying diagram, a ladder is leaning against a vertical wall, making an
angle of 70° with the ground and reaching a height of 10.39 feet on the wall.
Find, to the nearest foot, the length of the ladder.
Find, to the nearest foot, the distance from the base of the ladder to the wall.
Since the lines of sight are parallel, we conclude
the angle of elevation is congruent to angle of
depression by the alternate interior angles
theorem.
EX3: A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying
diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the
cliff is 18° and the angle of elevation to the top of the lighthouse is 28°. What is the height of the
lighthouse, x, to the nearest tenth of a foot?
EX4: You were flying a kite on a bluff, but you managed somehow to dump your kite into the lake
below. You know that you've given out 325 feet of string. A surveyor tells you that the angle of
declination from your position to the kite is 15°. How high is the bluff where you and the surveyor are
standing?