Happy 3 Day Week! Take Out: Applying Trig Notes. HW: Pg 310 # 29, 31, 37, 44, 47 Pg 316 #11,17, 32 Notecards: inverse trig functions, Law of Sines

AGENDA Review Unit 5 quiz 1 5.5: Inverse Trig Functions 5.6: Law of Sines 5.8: Law of Cosines ( if time) Whip Around 1st Period – Steph (U2T) 5th Period – Avalena (U2T) (U3Q1), Amanda (U2T) 6th Period –

Unit 5 Quiz 1 Students who still need to take it: Period 4: Melissa, Prerana 1st Period – Steph (U2T) 5th Period – Avalena (U2T) (U3Q1), Amanda (U2T) 6th Period –

1st Period – Steph (U2T) 5th Period – Avalena (U2T) (U3Q1), Amanda (U2T) 6th Period –

Learning Objectives By the end of this period you will be able to: Find missing angle measurements by using inverse trig functions. Use Law of Sines to solve triangles

Recap… Inverse Trig Functions: To find the angle measure arcsin(a) arccos (a) arctan (a) The expression sin-1 is read as “the inverse sine.” In this notation,-1 indicates the inverse of the sine function, NOT the reciprocal of the sine function. Reading Math

Whiteboards! Find all possible values of 225, 315 90, 270

Whiteboards! Find all possible values of cos-1 Find all possible values of tanθ= 1 30, 330 45, 225

Example 3

Whiteboards a. tan ( arcsin( 4/5) ) b. cos(arccos(2/5)) a)

Example 4 To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure.

Math Joke of the Day! Q: Why are you reading that sign backwards? A: It’s an inverse sine!

Whiteboard Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. DF= 5.7 <D: 68 <F: 22

Example 5 The rim of a basketball hoop is 10 feet above the ground. The free-throw line is 15 feet from the basket rim. If the eyes of a basketball player are 6 feet above the ground, what is the angle of elevation of the players line of sight when shooting a free throw to the rim of the basket?

Whiteboards! A plane is flying at an altitude of 12,000 m. The pilot knows that he went a horizontal distance of 19,204m. If the pilot is looking at the airport tower from the plane, what is the angle of depression?

Take out a piece of paper On the top, label it 5.6: Law of Sines and Law of Cosines. On the left hand corner, write U5L5.

So far, we have only use trig functions to calculate missing sides of right triangles. But what if we have an obtuse or an acute triangle? Today is your LUCKY day! We are going to use Law of Sines to calculate measures of other triangles.

Deriving Law of Sines You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In ∆ABC, let h represent the length of the altitude from C to AB.

You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) • two side lengths and a non-included angle measure

Practice with a Calculator: Whiteboards A. tan 103° B. cos 165° C. sin 93° It is okay if you get a negative number!

Example 1: Using Law of Sines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. FG FH 33.7

Example 2 Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mL 33.7

Whiteboard Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. MX 33.7

Whiteboard Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. NP 33.7

Whiteboard Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. AC 33.7

EX 3: Leaning Tower of Pisa! The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100°52’ angle with the ground. To stabilize the tower, an engineer considered attaching a cable that is 74.3m from the top of the tower to a point that is 40 m from the base. What angle does the cable make with the ground? Round to the nearest degree. CHANGE DIAGRAM!

Baseball! Ms. Huls went to a Giants game this past season. She sat directly behind home plate. The angle of depression to home plate is 29°54’ and the angle of depression to the pitcher’s mound is 24°12.’ In major league baseball, the distance between home plate and the pitcher’s mound is 60.5 feet. How far is Ms. Huls from home plate?

Law of Cosines The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines. REMEMBER: YOU CAN ONLY USE THE PYTHAGOREN THEOREM FOR RIGHT TRIANGLES.

Law of Cosines You can use the Law of Cosines to solve a triangle if you are given: • two side lengths and the included angle measure (SAS) • three side lengths (SSS).

Law of Cosines The angle referenced in the Law of Cosines is across the equal sign from its corresponding side. Helpful Hint Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator. Helpful Hint

Example 4: Using Law of Cosines Many times, you will have to use both Law of Sines and Law of Cosines to solve triangles. Solve the triangle. [ What do we need to solve for?] ( a) XZ (b) m X (c ) m Z

Whiteboards! Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ( a) m T (b) m S

Ex 5: Leaning Tower of Pisa..Again What if…? Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. 31 m

Whiteboards! A sailing club has planned a triangular racecourse, as shown in the diagram. How long is the leg of the race along BC? How many degrees must competitors turn at point C? Round the length to the nearest tenth and the angle measure to the nearest degree.

Whip Around We are going to share one thing we learned today. I will choose the first person to share, then you all will be responsible for choosing who will share next.