One More Week Until Break! Take Out: Unit Circle on yellow paper and notes from Monday. HW: Pg. 296 #1, 15-29 odd, #41, 43 Pg 302 # 23, 25 Pg 310 # 29, 31, 47 Notecards: angle of elevation and depression, inverse trig functions Updates: Unit 5 Quiz 1 (5.1-5.4) Thursday/Friday

AGENDA Notecard Practice Finish U5L3: Special Right Triangles and the Unit Circle Unit Circle Partners 5.4: Applying Trig Ratios Creation! 1st Period – Steph (U2T) 5th Period – Avalena (U2T) (U3Q1), Amanda (U2T) 6th Period –

Notecard Practice 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: Use special right triangles to derive the order pairs of the unit circle. Use trigonometry to find the measure of the sides or right triangles Find missing angle measurements by using inverse trig functions.

Unit Circle Unit Circle A circle with a radius of 1 unit. For every point P(x, y) on the unit circle, the value of r is 1. The coordinates of P can be written as (cosθ, sinθ) since the radius of a unit circle is 1.

Unit Circle On your yellow Unit Circle, trace over the y and x axis with a black pen. Label the degrees. ( this line divides the circle into fourths) Using a different color, trace the line that divides the circle into eighths. Label the degrees on the dotted line. Using a different color, trace the line that is closest to the x-axis. Label the degrees on the dotted line. Using a different color, trace the line that is closest to the y-axis. Label the degrees on the dotted line.

Unit Circle Now, lets write the ordered pairs for each terminal side. If a unit circle has a 1 unit radius what is an order pair for 0, 90, 180, 270?

Unit Circle You ONLY need to know the ordered pairs for the first quadrant, and then you can easily fill in the rest of the Unit Circle! Let’s look at the 30 degree ordered pair. If the radius ( or the hypotenuse was 1) what are the lengths of the other legs? Let’s use special right triangles

Unit Circle You ONLY need to know the ordered pairs for the first quadrant, and then you can easily fill in the rest of the Unit Circle! Let’s look at the 45 degree ordered pair. If the radius ( or the hypotenuse was 1) what are the lengths of the other legs? Let’s use special right triangles

Unit Circle You ONLY need to know the ordered pairs for the first quadrant, and then you can easily fill in the rest of the Unit Circle! Let’s look at the 60 degree ordered pair. If the radius ( or the hypotenuse was 1) what are the lengths of the other legs? Let’s use special right triangles

Unit Circle Now, with your table fill out the rest of the unit circle. Hint: All the green lines ( for example) should be the same ordered pair, but watch out for the negatives! Do this in pencil first. When you are confident, then raise your hand and write down the ordered pairs using the same color you drew the line.

Unit Circle A way to help you remember the ordered pairs is remember the lines closest to the x-axis ( same color) are the same ordered pairs and so on OR 3-2-1 1-2- 3 Look at the numerator of the x-coordinate.

Unit Circle On the Unit Circle, the following are true: Sinθ= y cosθ= x What about the other four trig ratios? Tanθ= cotθ= Secθ= Cscθ=

Example 2 Write this on your piece of binder paper. Use the Unit Circle to find each value. cos ( -180°) b) tan ( 270°)

Whiteboards! Use the Unit Circle to find each value. a) Sec ( 90°) b) cot ( 270°)

Example 3 Use the unit circle to find the values of the six trig functions for a 135° angle.

Whiteboards Use the unit circle to find the values of the six trig functions for a 210° angle.

Unit Circle Now highlight the following: 0 45 90 120 150 180 240 315 1 min

Unit Circle Partners It is time to interact with people that are NOT at your table!!! Find 9 different people. If you sign your name on someone’s paper for 45, then your partner MUST sign their name on 45 as well! Choose wisely. I might assign a partner quiz and you will have to do a quiz with one of your Unit Circle Partners. 1 min

Applying Trig Functions (5.4) Angle of Elevation: angle formed by a horizontal line and a line of sight to a point above the line. Angle of Depression: angle formed by a horizontal line and a line of sight to a point below the line. Angles of elevation and angle of depression are equal because they are alternate interior angles.

Whiteboards! Classify each angle as an angle of depression or elevation.

Solving Trig Applications Key Steps: Sketch a picture Figure out what you are looking for Solve with a trig ratio

Example 1(a) The Pilatusbahn in Switzerland is the world’s steepest cog railway. Its steepest section makes an angle of about 25°36’ with the horizontal and rises about 0.9 km. a)To the nearest hundredth of a kilometer, how long is this section of the railway track? b) What is the horizontal distance?

HELPFUL HINT When putting trig into your calculators, put everything in at once! When you round too early, you will be off.

Try and do 1(b) with your table-group! Example 1(b) The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 68°15’24’’ , how tall is the Space Needle? Round to the nearest meter. Try and do 1(b) with your table-group!

Whiteboards! A ladder is leaning against the side of a house so that the distance on the ground between the base of the ladder and the house is 7 feet. If the length of the ladder is 15 feet, what is the angle at which the ladder is leaning against the house? At what height does it reach the house?

Try and do 1(c) with your π Partner! Example 1(c) On May 18, 1980 Mount Saint Helen, a volcano in Washington, erupted with such force that the top of the mountain was blown off. To determine the new height of the summit of Mount Saint Helen, a surveyor measured the angle of elevation to the top of the volcano to 37° 46’. The surveyor than moved 1,000 feet closer to the volcano and measured the angle of elevation to be 40° 30’. Determine the new height of Mount Saint Helens. Try and do 1(c) with your π Partner!

Compare answers with your 240 Partner! Whiteboards! An observer in a lighthouse is 69 ft above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48º. The angle of depression to the other boat is 22º. What is the distance between the two boats? Round to the nearest foot. Compare answers with your 240 Partner!

Solving Right Triangles (5.5) Whiteboards: Find sin(30°) using the unit circle. But what if I just gave you the value and not the angle? This is called the inverse of sine or arcsine. Find Sinθ= ½ which can be written as sin-1 (1/2) or arcsin(1/2)

Solving Right Triangles (5.5) 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

Solving Right Triangles (5.5) Do not get mixed up with inverse trig functions and reciprocal trig functions. EX:

Whiteboards! Practice with our Calculators! Use your calculator to find each angle measure to the nearest degree. cos-1(0.87) . sin-1(0.85) tan-1(0.71) 29.5 58.2 35.37

Example 2

Example 3

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

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?

Creation! You have a quiz on Thursday/Friday. Write two trig application problems AND solve them. That contains regular trig ratios and/or elevation of depression/elevation Inverse trig I will use one of your questions on the quiz  If I choose yours, you will get 1 pt exta credit on the quiz so be creative!!!