Inverse Trigonometry sin −1 sin 𝐴 =𝐴 Recall that “inverse” in mathematics means the opposite. It is a function that undoes another function. The inverse of addition is subtraction 𝑎+4−4=𝑎 The inverse of multiplication is division 𝑏×4÷4=𝑏 The inverse of a square is a square root 𝑥 2 =𝑥 The inverse of a trig function is an inverse trig function sin −1 sin 𝐴 =𝐴

Andrea and Bonita are going for a walk straight up the side of a hill Andrea and Bonita are going for a walk straight up the side of a hill. Andrea decided to stretch before heading up the hill while Bonita thought this would be a good time to get a head start. Once Bonita was 100 feet away from Andrea, she stopped to take a break and looked at her GPS device that told her that she had walked 100 feet and had already increased her elevation by 40 feet. sin 𝐴 = 40 100 = 2 5 cos 𝐵 = 40 100 = 2 5

Inverse Trig Functions You can use inverse trigonometry to find the angle of elevation. sin 𝐴 = 40 100 = 2 5 sin −1 ( sin 𝐴) = sin −1 2 5 𝐴≈23.578° Bonita Andrea

Inverse Trig Functions Let’s use the same method to find angle B. cos 𝐵 = 2 5 cos −1 cos 𝐵 = cos −1 2 5 𝐵=66.4° Bonita Andrea

Find angle A sin −1 ( sin 𝐴 ) = sin −1 2 5 𝐴≈23.578° sin 𝐴 = 40 100 = 2 5 cos 𝐵 = 40 100 = 2 5 Find angle B cos −1 ( 𝑐𝑜s 𝐵 ) = cos −1 2 5 𝐵≈66.422°

Line of Sight Horizontal Line Angle of Depression Angle of Elevation Bonita Horizontal Line Can you see that the angle of elevation and the angle of depression are alternate interior angles? Andrea

Find all unknown values for the following triangle. ∠𝛼= ∠𝛽= ∠𝛾=90° 𝑎=12𝑚 𝑏=8𝑚 𝑐= =12 m =8 m The three angles in this triangle have Greek alphabet letters as variables. Alpha, Beta, and Gamma.

Represent a situation with a sketch Jill put a ladder up against the house to try and reach a light that is out and needs to be changed. She knows the ladder is 10 feet long and the distance from the base of the house to the bottom of the ladder is 4 feet. 10 feet 4 feet

Francis is a pilot of an airplane that if flying at an altitude of 3,000 feet when the plane begins its decent toward the ground. If the angle of decent of the plane is 15° how much farther will the plane fly before it is on the ground? 15° 3,000 feet

Abby is standing at the top of a very tall skyscraper and looking through a telescope at the scenery all around her. The angle of decline on the telescope says 35° and Abby knows she is 30 floors up and each floor is 15 feet tall. How far from the base of the building is the object that Abby is looking at? 35° 15∙30=450 450 feet

∠𝐵=90°−23.6 ∠𝐵=66.4° 𝐴𝐶= 10 2 − 4 2 𝐴𝐶= 84 =2 21 sin 𝐴 = 4 10 = 2 5 Find the missing sides and angles. sin 𝐴 = 4 10 = 2 5 sin −1 (sin 𝐴) = sin −1 2 5 ∠𝐴=23.6° ∠𝐵=90°−23.6 ∠𝐵=66.4° 𝐴𝐶= 10 2 − 4 2 𝐴𝐶= 84 =2 21

∠𝐴=90°−20 ∠𝐴=70° 𝐴𝐵= 82.4 2 − 30 2 𝐴𝐵≈76.8 tan 20 = 30 𝑎 𝑎 tan 20 =30 𝐴𝐵= 82.4 2 − 30 2 𝐴𝐵≈76.8

For each problem make a drawing, write an equation and solve. Carrie places a 10-foot ladder against a wall. If the ladder makes an angle of 65° with the level ground, how far up the wall is the top of the ladder? sin 65 = 𝑥 10 𝑥=10 sin 65 ≈9.06 feet x 10 feet 65°

A flagpole casts a shadow that is 15 feet long A flagpole casts a shadow that is 15 feet long. The angle of elevation at this time is 40°. How tall is the flagpole? sin 40 = 𝑥 15 15 sin 40 =𝑥 𝑥≈9.6 feet x 15 feet angle of elevation 40°

In southern California, there is a six mile section of interstate that increases 2,500 feet in elevation. What is the angle of elevation? There are 5,280 feet in a mile. 6 𝑚𝑖𝑙𝑒𝑠×5,280 𝑓𝑒𝑒𝑡 𝑚𝑖𝑙𝑒 =31,680 𝑓𝑒𝑒𝑡 tan 𝑥 = 2500 31680 = 125 1584 tan −1 tan 𝑥 = tan −1 125 1584 𝑥=4.512° 2500 feet 𝑥 6 miles

100 feet 35° x tan 35 = 100 𝑥 𝑥 tan 35 =100 𝑥= 100 tan 35 ≈142.8 feet A hot air balloon is 100 feet straight above where it is planning to land. Sarah is driving to meet the balloon when it lands. If the angle of elevation to the balloon is 35°, how far away is Sarah from where the balloon will land? tan 35 = 100 𝑥 𝑥 tan 35 =100 𝑥= 100 tan 35 ≈142.8 feet 100 feet 35° x

Determine the values of the two remaining trig ratios when given one of the trig ratios. cos 𝛼 = 3 5 Step one: Sketch the triangle Since cosine is adjacent leg over hypotenuse, 5 is the hypotenuse and 3 is the adjacent leg to ∠𝛼. Step two: Use the Pythagorean Theorem to find the other leg length. 5 2 − 3 2 =4 Step three: Write the remaining trig ratios. 3 5 𝛼 sin 𝛼 = 4 5 tan 𝛼 = 4 3