Algebra 2 / Trigonometry



Pre-College Algebra Name _____________________________

Date ________________ Period ________

Unit 1

Trigonometric

Functions

1-1 Trigonometry of the Right Triangle

1-2 Special Right Triangles

1-3 Radian Measure

1-4 Trigonometric Conversions

1-5 Reciprocal Trigonometric Functions

1-6 Finding Angles in Trigonometry on the Graphing Calculator

1-7 Trigonometric Co-functions

1-8 Identify amplitude, period and frequency from a trigonometric equation

Graphing the sine curve

1-9 Graphing the cosine curve

1-10 Graphing the tangent, secant, cosecant & cosecant curves

1-11 Writing a trigonometric equation given the curve

1-12 Vertical shifting of sinusoidal graphs

1-13 Rotations and Terminology

1-14 The Unit Circle

1-15 Definition of Sine, Cosine, Tangent on the Unit Circle

1-1 Trigonometry of the Right Triangle

Remember: SOH-CAH-TOA

Sine [pic] = measure of leg opposite the angle

measure of hypotenuse

Cosine [pic] = measure of leg adjacent to angle

measure of hypotenuse

Tangent [pic] = measure of leg opposite the angle

measure of leg adjacent to angle

1. Find the sine, cosine and tangent of the given angle.

a) sin A = b) cos A =

c) tan A = d) sin B =

e) cos B = f) tan B =

2. Use the diagrams to complete the following:

a) sin S = cos ____ = [pic]

tan ____ = [pic] sin ____ = [pic]

cos T = tan T =

b) sin X = cos X =

tan X = sin Z =

cos ____ = [pic] tan Z =

3 What is the value of [pic]?

|1) |[pic] |

|2) |[pic] |

|3) |[pic] |

|4) |[pic] |

4 By law, a wheelchair service ramp may be inclined no more than 4.76°. If the base of a ramp begins 15 feet from the base of a public building, which equation could be used to determine the maximum height, h, of the ramp where it reaches the building’s entrance?

|1) |[pic] |

|2) |[pic] |

|3) |[pic] |

|4) |[pic] |

5. If sin x = 3/5, find x to the nearest degree

6. If cos x = 2/3, find x to the nearest degree

7. If 2 tan x = 3, find x to the nearest degree

8 If tan x + 4 = -3.8, , find x to the nearest degree

9 If sin(x + 10) = 2/3, find x to the nearest tenth of a degree.

1-2 Special Right Triangles

Complete the table below.

| |[pic] |[pic] |[pic] |

|sin | | | |

|cos | | | |

|tan | | | |

Find the exact value of each of the following

1. cos 60o + sin 30o – tan 45o 2. sin 30o + cos 30o + tan 30o

3. (cos 60o)2 + (sin 30o)2 4. [pic]

5. [pic] 6. [pic]

7. Find the length of the altitude of an equilateral triangle whose side has a length of 6.

8. In the accompanying diagram, [pic] is an altitude of [pic]. If CD = 6, [pic] and [pic], find the perimeter of [pic] in simplest radical form.

9. Find the length of the altitude of an equilateral triangle whose side has a length of 10.

1-3 Radian Measure

Just as distance can be measure in inches, feet, miles, centimeters and so on, rotations about a point can also be measured in different ways. Measuring one complete rotation in terms of 360o is somewhat arbitrary.

A common unit of angle measurement that is an alternative to degrees is called the radian. It is defined in terms of the arc length of a circle and the circle’s radius.

[pic]

Radians measure the total number of radii that have been traversed about the circumference of a circle in a given rotation. Based on the circumference formula of a circle, there will always be [pic] radians in one full rotation.

Example: Find the arc length if the radius of a circle is 2 units and [pic] radian.

1. The distance from the center of a Ferris wheel to a person who is riding is 38 feet. What distance does a person travel if the Ferris wheel rotates through an angle of 4.25 radians?

(1) 80.75 feet (3) 507 feet

(2) 42.5 feet (4) 161.5 feet

2. If a pendulum swings through an angle of 0.55 radians, what distance does its tip travel if it has a length of 8 feet?

3. A dog is attached to a 10 foot leash. He travels around an arc that has a length of 25 feet. Which of the following represents the radian angle he has rotated through?

(1) 5 (3) 2.5

(3) 7.5[pic] (4) 1.25[pic]

4. A wheel whose diameter is 3 feet rolls a distance of 45 feet without slipping. Through what radian angle did the wheel rotate?

(1) 30 (3) 30[pic]

(2) 25 (4) 12[pic]

5. Determine the number of radians that the minute hand of a clock passes through if it has a length of 5 inches and its tip travels a total distance of 13 inches.

6. A golfer swings a club about a pivot point. If the head of the club travels a distance of 26 feet and rotates through an angle of 5 radians, which of the following gives the distance the club head is from the pivot point?

(1) 1.7 feet (3) 5.2 feet

(2) 2.6 feet (4) 7.2 feet

7. In a circle, a central angle of [pic] radians subtends an arc of 3 centimeters.  Find the length, in centimeters, of the radius of the circle.

1-4 Trigonometric Conversions

Relationship between Degrees and Radians:

Radians = degrees[pic] Degrees = radians[pic]

Convert each of the following to degrees or radians. Express answers in terms of [pic] when necessary.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

16. [pic] 17. [pic] 18. [pic]

19. [pic] 20. [pic] 21. [pic]

22. [pic] 23. [pic] 24. [pic]

25. [pic] 26. [pic] 27. [pic]

28. In a circle with a central angle of 60o, find the radius if the arc length is 4 feet.

29. In a circle with a central angle of 45o, find the arc length if the diameter is 10 feet.

30. In a circle with a central angle of 30o, find the arc length if the diameter of the circle is four feet.

31. If a circle has an arc length of 10 feet and a radius of 2 feet, find the amount of degrees in the

central angle to the nearest tenth.

32. If a circle has an arc length of 6 feet and a radius of 3, find how many radians the central angle is?

1-5 Reciprocal Trigonometric Functions

In a right triangle, there are actually six possible trigonometric ratios, or functions.

A Greek letter (such as theta [pic]or phi [pic]) will be used to represent the angle.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Notice that the three new ratios above are reciprocals of the ratios on the left. Applying a little algebra shows the connection between these functions.

[pic]

1. Given the triangle below, express the exact value of the six trigonometric functions in relation to theta.

[pic]

The following examples pertain to a right triangle in Quadrant I.

2. Given [pic], find csc [pic], [pic], tan [pic], sin [pic], cot [pic]

3. Find tan [pic], csc [pic],[pic] and [pic], given [pic] and [pic].

4. Determine the value of each of the following exactly in simplest form.

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i)[pic]

5. Use your calculator to determine the value of each of the following to the nearest hundredth.

(a) [pic] (b) [pic] (c) [pic]

6. Find the exact value of each of the following:

(a) [pic]   (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

7. In simplest radical form, [pic] is equal to

(1) [pic] (3) [pic]

(2) [pic] (4) [pic]

8. Which of the following is nearest to the value of [pic]?

(1) 1.19 (3) -2.74

(2) 3.17 (4) -0.85

1-6 Finding angles in Trigonometry on the graphing calculator

Degrees, Radians, and Trigonometric Angle Entries

The graphing calculator has 2 modes for angles – DEGREES and RADIANS. It is important to know which mode to use when working on a specific problem.

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Working with Degrees:

Set the MODE to Degree and all further calculations will be in degrees.

Working with Radians:

Set the MODE to Radian and all further calculations will be in radians.

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Degrees, Minutes, & Seconds

1. Convert 57° 45' 17'' to decimal degrees.

                                                                                                    

 

2. Convert 57° 45' 17'' to radians.

 

3. Convert 48.555° to degrees, minutes, seconds.

4. Find sin 57° 45' 17'':

5. Using your calculator, find the value of each of the following to three decimal places:

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] [pic] f) cot (3)

g) sec [pic] h) csc 47[pic] i) csc 23[pic]

j) tan[pic] k) csc [pic] l) sec [pic]

 

6. Given cos A = .0258.  Find ................
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