Section 6.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.

Transcription

1 1 Section 6.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint of the ray. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis. Positive angle a counterclockwise rotation Negative angle- a clockwise rotation. II. Degree 1 revolution = revolution revolution revolution

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3 Test One Notes 3 Coterminal Angles If angles α and β have the same initial and terminal sides, then they are coterminal and 3600 n, where n {, -2, -1, 0, 1, 2, } Example: Find a one positive and one negative coterminal angle to Complimentary: 900 Supplementary: 1800 α and β must be POSITIVE ANGLES Example: Find the compliment and the supplement of θ = 148 0

4 Test One Notes 4 III. Radians Measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. - Radian measure of an angle θ is the length of the arc that subtends the angle in a circle of radius 1. The Arclength (s) = radius (r) when θ = 1 radian, so: 1revolution 2 radian 1 revolution 2 1 revolution 4 radian 2 radian

5 Test One Notes 5 Example: Find the compliment of Example: Find a coterminal angle to Generally, α is coterminal with 2 n, n {, -2, -1, 0, 1, 2, }.

6 6 IV. Degrees and Radians conversions radians 1. Convert from degrees to radians: multiply the degrees by Convert from radians to degrees: multiply the radians by radians Example: Convert to radians. Find a negative coterminal angle in radians. Example: Convert 2 radians to degrees. Find a positive coterminal angle in degrees.

7 Test One Notes 7 V. Arc Length: s r, where θ is measured in RADIANS. What is the distance Sector a region bounded by 2 radii of the circle and their intercepted arc. Example: A sector of a circle has a central angle of Find the arc length if the radius is 15 inches. Example: The central angle θ in a circle of radius 5 meters is subtended by an arc of length 6 meters. Find the measure of angle θ in degrees and in radians.

8 8 1 2 VI. Area of a Sector of a Circle: A r, where θ is written in RADIANS! 2 Example: A car s rear windshield wiper rotates The wiper mechanism wipes the windshield over a distance of 14 inches. Find the area covered by the wiper mechanism.

9 Test One Notes 9 VII. Angular Speed ( ) How fast the angle changes when a particle moves along ( is swept out ) the circular arc of radius r at a constant speed. Think radians revolutions or time time Method I: Definition central _ angle time t, where θ is in RADIANS! Method II: Conversion # revolutions 2 radians time 1revolution Linear Speed ( ) Measures how fast a particle moves along the circular arc of radius r. Movement is assumed to be a constant value/speed. Think length time miles hour cm sec Method I: Formula arc _ length time s t r t, where θ is in RADIANS! Method II: If you know the angular speed, use r

10 10 Example: My truck tires have a 20 radius and turn at 4 revolutions per second. How fast is my truck moving? (Write the final answer in miles per hour). Example: A Ferris wheel with a 100 foot diameter makes 1.5 revolutions per minute. A) Find the angular speed of the Ferris wheel in radians per minute. B) Find the linear speed of the Ferris wheel.

11 Test One Notes 11 Section 6.2 Right Triangles Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and r = x2 y / 0 For 0 2 : sin opp hypot y r csc hypot opp r y cos adj hypot x r se c hypot adj r x tan opp adj cot adj opp y x Example: Triangles x y

12 12 Reciprocal Identities Quotient Identities sin 1 csc csc 1 sin tan sin cos cos 1 sec sec 1 cos cot cos sin tan 1 cot cot 1 tan Example: Triangle

13 Test One Notes 13!!!!The Unit Circle Memorize this page!!!! θ in radians θ in degrees cos θ sin θ tan θ

14 14 Angles of Elevation and Depression Angle of Elevation the acute angle from the horizontal up to the line of sight of the object. Angle of Depression the acute angle from the horizontal down to the line of sight of the object. Example: A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 35,000 ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is Draw the picture for this setting. You will answer questions regarding this picture in the homework. Example: From the top of a 200- ft lighthouse, the angle of depression to a ship in 0 the ocean is 23. How far is the ship from the base of the lighthouse?

15 15 Comment: An angle of 0 with the ground means to create this picture: Example: An 930 inch guy wire is attached to the top of a tower, making a 65 0 angle with the ground. How high is the tower in feet? Example: A woman standing on a hill sees a flagpole that she knows is 60 feet tall. The angle of depression to the bottom of the pole is 14 0, and the angle of elevation to the top of the pole is Find her distance x from the pole.

16 16 Example (time permitting): The Freedom Tower is 1776 feet tall. The angle of elevation from the base of an office building to the top of the tower is The angle of elevation from the roof of the office building to the top of the tower is A) How far is the office building from the Freedom Tower, measured to the nearest foot? B) How tall is the office building, to the nearest foot?

17 Test One Notes 17 Section 6.3 Reference Angle Let θ be an angle in standard position. It s reference angle is the acute angle formed by the terminal side of θ and the horizontal axis. Quadrant I Quadrant II Quadrant III Example: Find the reference angle ' for each angle below: a) θ=300 0 b) θ= 2 3 c) θ=6 d) θ=-7 Quadrant IV

18 Test One Notes 18 Trigonometric functions for any angle (not just acute angles) Method I: Use the unit circle and a reference angle. 1. Determine the quadrant that θ lies in. 2. Find the reference angle θ for θ 3. Find the values of the trigonometric functions for the reference angle: sin θ cos θ tan θ csc θ sec θ cot θ 4. Assign the appropriate signs. sin θ=±sin θ cos θ=±cos θ Example: Calculate a) Cot(-135 ) 0 19 b) Sec( 6 ) tan θ=±tan θ

19 Test One Notes 19 Method II: Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and r = x2 y Determine the quadrant that θ lies in. 2. Draw your triangle. 3. Use the Pythagorean theorem to find the missing side length. 4. Find the values of the trigonometric functions. 5. Attach the correct sign. Practice for step 1, determine the quadrant. Ex) From the information given, find the quadrant in which the terminal point determined by theta lies. a) sec θ>0, tan θ < 0 b) tan θ < 0, sin θ < 0 d) sec 5 and sinθ < 0 Ex) Find the sign of the expression if the terminal point determined by a) sin csc, b) tan sin, cos is in Quadrant II is in Quadrant III is in the given quadrant.

20 20 Ex) Calculate csc θ when cos 2 7 and θ is in Quadrant III. Ex (if time): Let (-3, 4) be a point on the terminal side of θ. Find the exact value of the six trigonometric functions.

21 21 Example (if time): Let tanθ = 5 4 and cosθ > 0. Find the exact value of the six trigonometric functions. Ex: Find the values of the trigonometric functions of when sec =-3, and the terminal point of is in Quadrant III.

22 22 Pythagorean Identities 2 2 sin cos tan 1 sec 2 2 cot 1 csc Proof: Example: Let θ be an acute angle such that sinθ =.6. Find the cosθ and tanθ using only identities.

23 23 Ex: Write the first expression in terms of the second if the terminal point determined by the given quadrant. is in a) tan in terms of sin, where is in Quadrant III. b) 2 2 sec sin in terms of cos( ), for any quadrant. c) sin t in terms of sect; Quadrant IV

24 24 Area of a SAS Triangle: 1 K absin C 2 The area of a SAS triangle is half the product of the two sides times the sine of the included angle. Example: Find the area of the shaded segment of a circle whose radius is 8 feet, formed by a central angle of Example (time permitting): A parking lot has the shape of a parallelogram. The lengths of the two adjacent sides are 70m and 100m. The angle between the two sides is What is the area of the parking lot?

25 25 Section 7.1 The Unit Circle Rule 1: The terminal point on the unit circle can be written as P(x,y). Rule 2: The terminal point is determined by a real number t Comment:This is really an angle measure, and anytime you see in the notes, it is the same as t. Ex) Find the terminal point P(x, y) on the unit circle determined by the given value of t= Ex) Find the terminal point P(x, y) on the unit circle determined by the given value of t= 7 2.

26 26 Rule : Points on the unit circle satisfy the equation x y Ex) Show that the point is equation 2 2 x y , 7 7 on the unit circle by verifying that it satisfies the Ex) The point P is on the unit circle. Find P(x, y) from the given information. a) The y-coordinate of P is 1 and the x-coordinate is positive. 3 b) The x-coordinate of P is 3 5 and P lies in quadrant III.

27 The Unit Circle Quadrant Angles Comment: The hw and text will be using t Ex: Find the value of each of the six trigonometric functions (if it is defined) at the given real number t. Use your answers to complete the table. (If an answer is undefined, enter UNDEFINED.) a) b) t = 9 2

28 28 Even Odd Properties Example 0 a) cos 270 b) sin 3 2 c) tan 6

29 29 Using the Calculator A calculator will give approximate values (not exact) for the trigonometric functions. 1. Choose Mode. Choose degrees if there is a 0 symbol. Choose radians if there is no symbol. 2. Sinθ, cosθ, and tanθ are on the calculator. 3. For cscθ, type in 1/sinθ For secθ, type in 1/cosθ For cotθ, type in 1/tanθ Ex) Use the calculator to find the following. Round your answer to six decimal places. a) sin(2.2) b) cot(28) c) csc(0.98) d) sec(5)