1. Trigonometry Survival Manual
Lesson 9-T
Trigonometry Survival Manual

2. Objectives
Go over the basics of Trigonometry

3. Vocabulary Cosine, abbreviated cos Sine, abbreviated sin
Tangent, abbreviated tan

4. Trigonometric Functions
Sin (angle) = Opposite / Hypotenuse
Cos (angle) = Adjacent / Hypotenuse
Tan (angle) = Opposite / Adjacent
SOH – CAH – TOA (or others) to help remember the definitions
Some Old Hillbilly Caught Another Hillbilly Throwing Old Apples
hypotenuse
opposite
angle
adjacent

5. Trig Relationship between two acute angles
Sin A is opposite over hypotenuse: a/c
Cos A is adjacent over hypotenuse: b/c
Tan A is opposite over adjacent: a/b a b c A B C
Sin B is opposite over hypotenuse: b/c
Cos B is adjacent over hypotenuse: a/c
Tan B is opposite over adjacent: b/a
So, Sin A = Cos B and Cos A = Sin B
a2 + b2 = c2 (from Pythagorean Theorem)
mA + mB = 90° (3’s of ∆ = 180°)

6. Trig Problems Steps to Solution
Step 1: Label sides (A, H, O) based on angle
Step 2: Identify trig function to use
Step 3: Set up equation
Step 4: Solve for variable (1 of these methods)
if variable is in top of fraction, multiply both sides by the bottom to get “x = …”
if variable is in bottom of fraction, x trades places with what’s on the other side of the = sign to get “x = …”
if variable is the angle, use inverse trig function notation to get “x = …” x
sin 23° = x = 45  sin 23° 45
cos 41° = x =
x cos 41°
tan x° = x = tan

7. Example 1: (variable on top)
16 is H , x is A and no value for O
Since we have A and H we need to use cos x
cos (37°) =
16 cos (37°) = x = 12.78
(x is on top multiply both sides by bottom) x
37° 16 y° H O
Use 90 – 37 = 53 to find the other angle, y A

8. Example 2: (variable on bottom)
19 is O , x is A and no value for H
Since we have O and A (no H) we need to use tan
tan (42°) = x
x = = tan (42°)
(x is on bottom then switch it with the trig function) x 19
42° y° O H A
Use 90 – 37 = 53 to find the other angle, y

9. Example 3: (variable is angle)
12 is H , 8 is O and no value for A -- x is the angle !
Since we have O and H we need to use sin
sin (x°) = (x is angle use inverse sin)
x = sin-1 (8/12) = 48.19° x° 12 8 H
Use Pythagorean Theorem to find one missing side
12² = z² + 8²  = z²  = z²  = z O z A

10. Angles of Elevation or Depression
To Solve:
Step 1: Draw the triangle below
Step 2: Label sides (A, H, O) from problem information
Step 3: Identify trig function to use
Step 4: Set up equation
Step 5: Solve for variable (use 1 of the 3 methods)
slant distance;
Ladders, ski slope or road
vertical distance
or height
angle always goes here Θ
Ground or horizontal distance
or length of shadow

11. Example 1: (variable on bottom)
The bottom of the board leaning up against a barn is 8 feet away from the side of the barn. If the board forms a 54° angle with the ground, how long is the board?
Draw and label the triangle: x (H) is the length of the board and it is 8 feet away (A) (along the ground) from the vertical side of the barn (O).
Since we have A and H, we need to use cos
cos 54° = 8/x [variable on bottom]
x = 8/(cos 54°) = feet H x O
angle
54° A 8
Since x is on bottom, we switch the variable and the trig function

12. Example 2: (variable on top)
A person walks 30 feet from the base of a tree and measures the angle to the top of the tree as 38°. How tall is the tree?
Draw and label the triangle: x (O) is the height of the tree. 30 feet away (A) (along the ground) from tree the angle is measured. The slant distance (H) is unknown and not needed.
Since we have O and A, we need to use tan
tan 38° = x/ [variable on top]
30 tan 38° = x = foot tall tree H x O
angle
38° A 30
Since x is on top, we multiply both sides by the bottom

13. Example 3: (variable is angle)
What angle is formed between a 18 foot ladder and the floor, if the end of the ladder is 10 feet up the side of the gym? 18
Draw and label the triangle: 18 (H) is the length of the ladder and it is 10 feet up (O) the vertical wall of the gym. x is the angle the ladder forms with the floor and (A) is not given.
Since we have O and H, we need to use sin
sin x° = 10/ [variable is angle]
sin-1 (10/18) = x = 33.75° H 10
angle O x° A
Since x is the angle, we need to use inverse trig function, 2nd Key then trig function