9.1 – Trigonometric Ratios (PART 1)

Why is Trigonometry important? Flight of Planes Sound Waves (study in music theory classes) Current of the Ocean Architecture Navigation/Surveying Without Trig. we would have NEVER made it to the moon! Launch and Orbital Trajectories

Topic One Solving for missing pieces of information.

Right Triangle Trig. Functions Let  be an acute angle of a right triangle. sin 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos 𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tan 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 SOH CAH TOA

Identifying the correct angle makes a difference!

Evaluate the Trig. Functions for 𝜃 sin 𝜃 = 15 𝜃 hypotenuse sin 𝜃 = 15 17 17 8 cos 𝜃 = adjacent cos 𝜃 = 8 17 tan 𝜃 = opposite ta𝑛 𝜃 = 15 8

Evaluate the Trig. Functions for 𝜃 hypotenuse sin 𝜃 = 14 𝜃 sin 𝜃 = 23 29 145 5 29 23 opposite cos 𝜃 = cos 𝜃 = 14 29 145 tan 𝜃 = adjacent ta𝑛 𝜃 = 23 14

THIS IS WHEN THE MODE OF YOU CALCULATOR MATTERS!!!! Find the Value of x THIS IS WHEN THE MODE OF YOU CALCULATOR MATTERS!!!! When you evaluate trig. functions at angles you need to be in the correct angle units. sin 47 = 𝑥 15 15 x 47 hypotenuse 15 sin 47 =𝑥 𝟏𝟎.𝟗𝟕=𝒙 opposite

Find the Value of x cos 31 = 15 𝑥 x cos 31 =15 x= 15 cos 31 𝒙=𝟏𝟕.𝟓𝟎 x 31 hypotenuse x cos 31 =15 x= 15 cos 31 adjacent 𝒙=𝟏𝟕.𝟓𝟎

Topic Two Solving for ALL of the missing pieces of information.

Solving Right Triangles When asked to SOLVE A RIGHT TRIANGLE Find ALL missing sides Find ALL missing angles What to Use when solving a right triangle: SOH – CAH – TOA Pythagorean Thm. Triangle Sum Thm. Inverse Trig (solve for unknown angles)

Pythagorean Thm. 𝑙𝑒𝑔 2 + 𝑙𝑒𝑔 2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 2 Triangle Sum Theorem All angles in a triangle add = 180∘

Change your calc to radian mode and see what happens… Inverse Trig – to find UNKNOWN angles!!! 𝜃= sin −1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝜃= cos −1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝜃= tan −1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 15 𝜃 18 𝜃= tan −1 18 15 𝜃= tan −1 1.2 𝜽=𝟓𝟎.𝟏𝟗∘ Change your calc to radian mode and see what happens… 𝜃= tan −1 1.2 𝜽=.𝟖𝟕𝟔

Solve the Right Triangle 55∘ 10.65 𝒄=𝟕.𝟒𝟔 Side c: sin 35 = 𝑐 13 OR 10.65 2 + 𝑐 2 = 13 2 113.42+ 𝑐 2 =169 𝑐 2 =55.58 𝒄=𝟕.𝟒𝟔 Side a: cos 35 = 𝑎 13 13cos 35 =𝑎 𝒂=𝟏𝟎.𝟔𝟓 𝑚∠𝐴=180−90−35 𝒎∠𝑨=𝟓𝟓∘

Solve the Right Triangle 𝟐𝟐.𝟔𝟐∘ 𝟔𝟕.𝟑𝟖∘ 𝟏𝟐 Find side e: (Pythagorean triple!) 5 2 + 𝑒 2 = 13 2 25+ 𝑒 2 =169 𝑒 2 =144 e=𝟏𝟐 sin ∠𝐷 = 5 13 ∠𝐷= sin −1 5 13 ∠𝑫=𝟐𝟐.𝟔𝟐∘ 𝑚∠𝐸=180−90−22.62 𝒎∠𝑬=𝟔𝟕.𝟑𝟖∘

Topic Three Angles of Elevation and Depression

Angle of Elevation & Depression

Angle of Elevation An ant looks up at the top of a tree with an angle of elevation of 62∘. (The ants height is negligible). If the tree is 15 feet tall, how far away is the ant from the top of the tree?

Angle of Depression From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is 34º. How far is the object from the base of the cliff?

Homework Topic One: Textbook Pg. 333#5, 6 Topic Two: Textbook Pg. 334 #1, 2, 8 Topic Three: Workbook Pg. 67 #16-18