1. Chapter 6: Trigonometry 6.2: Trigonometric Applications
Essential Question: What does SOHCAHTOA mean and how is it used to solve a right triangle problem?

2. 6.2: Trigonometric Applications
Old stuff will be used in this section
Triangle Sum Theorem
The sum of the measures of the angles in a triangle is 180°
Pythagorean Theorem
In a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2

3. 6.2: Trigonometric Applications
Finding a side of a Triangle
Find side x in the right triangle below
In this figure, we’re given: An angle (65°) The Hypotenuse (8) A side Adjacent to 65° (x) The sides we’re using are A and H using SOH-CAH-TOA means we use the cosine function 8
65° x

4. 6.2: Trigonometric Applications
Finding an Angle of a Triangle
Find the measure of the angle θ in the triangle below In this triangle, we’re given all three side lengths, so we can use any of the trigonometric ratios to solve.
SOH
sin θ = 3/5 → sin-1(3/5) = °
CAH
cos θ = 4/5 → cos-1(4/5) = °
TOA
tan θ = 3/4 → tan-1(3/4) = °
All ratios give us the same answer: ° 4 3 θ 5

5. 6.2: Trigonometric Applications
Solving a Right Triangle
Solve the right triangle below
The Triangle Sum Theorem helps find θ 75° + θ + 90° = 180° θ = 15° We can use the hypotenuse (17) and the 75° angle to find sides a and b θ 17 a
75° b

6. 6.2: Trigonometric Applications
Solving a Right Triangle
Solve the right triangle below
The Pythagorean Theorem helps find a a = a2 = a = We can find β by using the cosine function cos β = 6/12 cos β = 1/2 cos-1(1/2) = β 60° = β We can either find θ by using the sin function or by using The Triangle Sum Theorem θ = 30° θ 12 a β 6

7. 6.2: Trigonometric Applications
Assignment
Page 429
Problems 1 – 35, odd problems
Questions where you’re told to not use a calculator can be solved using the chart you copied on Tuesday.

8. Chapter 6: Trigonometry 6.2: Trigonometric Applications Day 2
Essential Question: What does SOHCAHTOA mean and how is it used to solve a right triangle problem?

9. 6.2: Trigonometric Applications
A straight road leads from an ocean beach at a constant upward angle of 3°. How high above sea level is the road at a point 1 mile from the beach?
Answer
If one is not drawn, DRAW A DIAGRAM.
Looking for the side on the right of the triangle, which is the side opposite of 3°
sin 3° = x/ • sin 3° = x ft = x
5280 ft 3° h

10. 6.2: Trigonometric Applications
According to the safety sticker on a 20-foot ladder, the distance from the bottom of the ladder to the base of the wall on which it leans should be one-fourth of the length of the ladder: 5 feet.
How high up the wall will the ladder reach
If the ladder is in this position, what angle does it make with the ground?
Draw a diagram
ladder 20 ft
wall h ft θ
ground 5 ft

11. 6.2: Trigonometric Applications
The wall height can be found using the Pythagorean Theorem
52 + h2 = 202
h2 = 400 – 25
h2 = 375
h = (375)½ ≈ ft
We’re given the side adjacent to θ and the hypotenuse, meaning we need to use cosine
cos θ = 5/20
θ = cos-1(5/20) ≈ 75.5°
ladder 20 ft
wall h ft θ
ground 5 ft

12. 6.2: Trigonometric Applications
Angles of Elevation and Depression
Both create right angles from an endpoint. Angles of elevation look up; angles of depression look down.
Angle of elevation
Angle of depression

13. 6.2: Trigonometric Applications
Elevation/Depression
A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole.
Draw a diagram
You’re given a 35° angle
You’re given the side adjacent
You’re looking for the side opposite
You’re using tangent
tan 35° = x / 60
60 • tan 35° = x
≈ x
flagpole h ft
35°
shadow 60 ft

14. 6.2: Trigonometric Applications
Elevation/Depression (#3: both)
A person on the edge of a canal observes a lamp post on the other side with an angle of elevation of 12° to the top of the lamp post and an angle of depression of 7° to the bottom of the lamp post from eye level. The person’s eye level is 152 cm.
Find the width of the canal.
Find the height of the lamp post.
Draw a diagram
top half of lamp post
12°
canal 7°
152 cm
152 cm

15. 6.2: Trigonometric Applications
Elevation/Depression (#3: both)
The canal is adjacent to the 7° angle
You’re given 152 cm, which is opposite 7°
Use tangent
tan 7° = 152 / x
x = 152 / tan 7°
x = cm
Use the canal measurement to find the top half of the lamp post… again using tangent.
tan 12° = y /
• tan 12° = y
cm = y
So the height of the lamp post is = cm
top half of lamp post (y)
12°
canal (x) 7°
152 cm

16. 6.2: Trigonometric Applications
Assignment
Page 431
Problems 37 – 49, odd problems
Quiz tomorrow
DMS/decimal conversion
Finding 6 trig ratios
Solving right triangles
A word problem or two