Trigonometry Survival Manual Lesson 9-T Trigonometry Survival Manual

Objectives Go over the basics of Trigonometry

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

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

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°)

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 21 21 cos 41° = ------- x = --------- x cos 41° 23 23 tan x° = ------- x = tan -1 ----- 37 37

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 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

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 19 tan (42°) = ----- x 19 x = ------------ = 21.10 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

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 8 sin (x°) = ------ (x is angle use inverse sin) 12 x = sin-1 (8/12) = 48.19° x° 12 8 H Use Pythagorean Theorem to find one missing side 12² = z² + 8²  144 = z² + 64  80 = z²  8.94 = z O z A

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

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°) = 13.61 feet H x O angle 54° A 8 Since x is on bottom, we switch the variable and the trig function

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/30 [variable on top] 30 tan 38° = x = 23.44 foot tall tree H x O angle 38° A 30 Since x is on top, we multiply both sides by the bottom

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° = 10/18 [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