DAY 1 - GEOMETRY FLASHBACK

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2 DAY 1 - GEOMETRY FLASHBACK Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse sin θ = opp. hyp. cos θ = adj. hyp. tan θ = opp. adj. Tangent Opposite Adjacent a 2 + b 2 = c 2 csc θ = hyp. opp. sec θ = hyp. adj. cot θ = adj. opp.

3 DAY 1 - PRACTICE Find the exact values of the six trigonometric functions of the angle given in the figure. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

4 DAY 1 - PRACTICE Find the exact values of the six trigonometric functions of the angle given in the figure. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

5 DAY 1 - PRACTICE Find the exact values of the six trigonometric functions of the angle given in the figure. cos θ = 3 7 sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

6 DAY 1 - PRACTICE Use the given function value(s) and the trigonometric identities to find the indicated trigonometric functions. Given: sec θ = 5 Find: sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

7 DAY 1 - PRACTICE Use the given function value(s) and the trigonometric identities to find the indicated trigonometric functions. Given: Find: sin θ = 1 2 csc θ = cos θ = cos(90 θ) = tan θ = cot(90 θ) =

8 HOMEWORK (DAY 1) p. 310 #1 23 odd

9 WARM UP (DAY 2) Find the six trigonometric ratios. cot θ = 2 6 sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

10 DAY 2 - ANGLES An angle is a rotation measure between two rays Angles in a triangle are measured in degrees and can be broken into smaller increments (minutes & seconds) 1 = 60 (1 degree = 60 minutes) 1 = 60 (1 minute = 60 seconds) = 42 degrees, 10 minutes, and 15 seconds To convert into a decimal equivalent use one of two methods Calculator (2 nd ANGLE) By Hand degree + minutes 60 + seconds 3600

11 DAY 2 - FINDING ANGLES AND SIDES PRACTICE Use a calculator to evaluate the following. (Round answers to the nearest ten thousandths.) 1) sin 30 4) csc 278 2) cos ) sec ) tan " 6) cot " Use a calculator to find the following angles (0 θ 90 ) 1) sin θ = 1 2 4) csc θ = 3 2) cos α = 2 2 5) sec θ = ) tan β = ) cot θ =

12 DAY 2 - SOLVING RIGHT TRIANGLES Before you can solve a triangle, you must know how to find missing sides and angles. 1)Make sure the calculator is in degree mode. 2)Determine what you are given and what you are looking for (angle or side). Given Angle Side Use sin θ, cos θ, or tan θ sin 1 (x), cos 1 (x), or tan 1 (x) There is a difference in sin 1 (x) and (sin θ) 1

13 DAY 2 - SOLVING RIGHT TRIANGLES Find the missing side (to the nearest tenth) or angle (to the nearest second) of the given triangles. Draw and label all three sides and angles of a right triangle that has a 40 angle and a hypotenuse of 10 cm.

14 HOMEWORK (DAY 2) p. 311 #39 61 odd

15 WARM UP (DAY 3) Evaluate: 1. cos( ") = 2. csc = Find the value of θ: 1. sinθ = cotθ =

16 DAY 3 - SOLVING RIGHT TRIANGLE WORD PROBLEMS 1) A 6-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is 132 feet from the tower and 3 feet from the tip of the tower s shadow, the person s shadow starts to appear beyond the towers shadow. What is the height of the tower?

17 DAY 3 - SOLVING RIGHT TRIANGLE WORD PROBLEMS 2) A biologist wants to know the width of a north-south river in order to properly set instruments for studying pollutants in the water. From point A, the biologist looks west, across the river, to Point C. Then walks south 100 feet (to point B) and again sights to point C. From this sighting it is determined that ABC = 58. How wide is the river?

18 DAY 3 - ANGLE OF ELEVATION & DEPRESSION Angle of Elevation from the horizontal up Angle of Depression from the horizontal down 3) From a 60-foot observation tower on the coast, a Coast Guard officer sites a boat in difficulty. The angle of depression of the boat is How far is the boat from the tower?

19 DAY 3 - ANGLE OF ELEVATION & DEPRESSION 4) The track of the Duquesne Incline in Pittsburgh, PA is about 800 feet long and the angle of elevation is 30. The average speed of the cable cars is about 320 feet per minute. a) How high does the Duquesne Incline rise? b) What is the vertical speed of the cable cars (in feet per minute)?

20 HOMEWORK (DAY 3) p #1, 7, 11, 13 19, 21, 24, 27, 37, 39, 49

21 DAY 4 - RIGHT TRIANGLE WORD PROBLEMS 1) A ship sailing directly towards an observer at the top of a 2000 feet tall cliff. At 4:00 the observed angle of depression is 8. Ten minutes later, the observed angle of depression is 37. What is the approximate speed of the ship in miles per hour?

22 DAY 4 - RIGHT TRIANGLE WORD PROBLEMS 2) The angle of elevation from a point on level ground to the top of a tower is The angle of elevation of the top of the tower from a point 145 feet further back on the same horizontal line is Find the height of the tower to the nearest tenth of a foot.

23 DAY 4 - RIGHT TRIANGLE WORD PROBLEMS 3) You are looking down from the top of a building 630 feet tall. As you look down at a nearby smaller building, you observe that the angle of depression to the base is 80 and the angle of depression to the top is 27. Find the height of the smaller building and the horizontal distance between the buildings.

24 DAY 4 - RIGHT TRIANGLE WORD PROBLEMS 4) Find the value of x and y as indicated in the diagram.

25 HOMEWORK (DAY 4) Finish word problems on hand out

26 EXTRA PRACTICE (IF NEEDED)

27 WARM-UP (DAY 5)

28 DAY 5 - DEFINITIONS Angle determined by rotating a ray about its endpoint Initial side starting position of the ray Terminal side ending position of the ray Vertex endpoint of the ray vertex initial side

29 DAY 5 - DEFINITIONS Standard position initial side is the positive x-axis and the origin is the vertex Positive angles counterclockwise rotation Negative angles clockwise rotation

30 DAY 5 - DEFINITIONS Quadrantal angles an angle whose terminal side does not lie in a quadrant Complementary angles angles whose sum is 90 Supplementary angles angles whose sum is 180

31 DAY 5 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Given an ordered pair on the terminal side of an angle in standard position, the six trigonometric functions can be found using right triangle trigonometry To find r, use the Pythagorean Theorem with x & y as the legs of the triangle

32 DAY 5 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE sin θ = y r csc θ = r y *Reminder x 2 + y 2 = r 2 cos θ = x r tan θ = y x sec θ = r x cot θ = x y

33 DAY 5 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Find the six trigonometric functions of an angle whose terminal side passes through the point (12, 16).

34 DAY 5 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Find the six trigonometric functions of an angle whose terminal side passes through the point ( 9, 40).

35 DAY 5 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Find the six trigonometric functions of an angle whose terminal side passes through the point ( 60, 25).

36 DAY 5 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Find the six trigonometric functions of an angle whose terminal side passes through the point (2, 4).

37 HOMEWORK (DAY 5) p. 320 #1-12 all TEST FRIDAY, FEBRUARY 2 nd, 2018

38 WARM-UP (DAY 6) Find the six trig functions whose terminal side passes through the given point. 1) (3, 1) 2) 2, 4 3) ( 5, 2 5) 4) 1, 1

39 DAY 6 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE The trigonometric functions are periodic and therefore follow a pattern and repeat Look at the previous four problems and determine in which quadrant each function is positive.

40 DAY 6 - TRIG PRACTICE OF ANY ANGLE Find the values of the six trig functions of θ. 1) sin θ = 5 13 θ lies in quadrant II 2) cos θ = & cot θ > 0 3) tan θ = & csc θ < 0

41 DAY 6 - TRIGONOMETRIC FUNCTIONS OF ANY ANGLE List the quadrant in which each scenario occurs 1. cos θ > 0 & tan θ < 0 2. sin β < 0 & sec β < 0 3. cot α > 0 & tan α < 0 4. csc γ > 0 & cos γ > 0 5. sec μ < 0 & sin μ > 0

42 HOMEWORK (DAY 6) p #13-18 all #19-29 odd # all TEST FRIDAY, FEBRUARY 2 nd, 2018