Reciprocal Identities Quotient Identities Pythagorean Identities

Transcription

1 2 Precalculus Review Sheet Fundamental Identities: Reciprocal Identities Quotient Identities Pythagorean Identities = csc! cos! = tan! sin2! + cos 2! = cos! = sec! cos! = cot! tan2! + = sec 2! tan! = cot! + cot2! = cs c 2! Even/Odd Identities sin(!") =!sin" cos(!") = cos" tan(!") =! tan" (0, ) (-, 0) (, 0) Cofunction Identities = cos( 90! "!) sec! = csc( 90! "!) tan! = cot( 90! "!) (0, -) Test Objectives: I. You should be able to sketch a reference triangle for a given angle in degrees or radians, and evaluate trig functions exactly without the use of a graphing calculator in any quadrant.. cos csc tan 20 " 4. sin 5! # 6 % & ' 5. cot " 4! % # 3 & ' 6. sec " 7! # 4 % & ' II. You should be able to evaluate any trig function using your calculator in the correct mode. Round to four decimal places. 7. sin cos 5 rad 9. tan (-4.78) 0. cot 92. csc sec 7

2 III. You should be able to give the correct reference angle for a given angle. If the angle is given in radians, give the reference angle in radians. If the angle is given in degrees, give the reference angle in degrees ! 3 " ! 0 IV. Given the trig function of an angle and the quadrant where the angle terminates OR a point on the terminal side of an angle, you should be able to determine the angle and the remaining trig functions. 7. Given cos! = -/2 and! is in Quadrant II, determine the remaining five trig functions. 8. Given csc! = - 2 and! terminates in Quadrant IV, determine sin!, cos! and tan!. 9. Given cot! = 3 and! terminates in Quadrant III, determine the remaining five trig functions. 20. Given tan! =! 2 3 and cos! > 0, determine csc!. 2. Given sec! = -3 and tan! > 0, determine cot!. 22. Given sin! =!2 7 and cot! < 0, determine sec!.

3 V. Given a point on the terminal side of an angle and the quadrant where the terminal side is located, you should be able to determine the EXACT values of the other trig functions. (EXACT means to include a radical, if appropriate, in your answer do NOT approximate with your calculator.) 23. The point ( 3,! 5 ) is on the terminal side of angle!. Find the exact value of the all of the trig functions of!. 24. The point (!4, 7 ) is on the terminal side of angle!. Find the exact value of all the trig functions of!. VI. Given the trig function of a multiple of one of the special angles OR quadrantal angles, you should be able to solve for all possible angles for a given domain. Solve for! in the following equations over the domain values: 0! " < 360! 25. sin! = cos! = sin! = ½ 28. csc! =! tan! = sec! = 2 3 Solve for! in the following equations over the domain values: 0! " < 2# 3. sec! = sin! = cot! = 3 VII. You should know the domain, range and period for the sine and cosine functions. 34. y = 35. y = cos! Domain: Range: Period: Domain: Range: Period:

4 VIII. You should be able to use triangle trigonometry to solve application problems. Quick Review activity on line to get your thinking fast The angle of elevation from an observer standing 50 feet from a building is 78 degrees. How tall is the building? 37. A monkey seated on a branch 2 feet above the ground peers down through an angle of depression of 0 degrees to a banana on the ground. If he drops directly to the ground, how far will he run to get to the banana? 38. A little boy lets out exactly 00 feet of kite string and looks up at his kite through an angle of elevation of 35 degrees. How high is the kite? 39. As you drive from the airport in Denver towards the mountains in the distance, you first see the top of a mountain through an angle of elevation of approximately 5.7 degrees. After traveling 0 miles, the mountain top is now visible through an angle of elevation of approximately 6.7 degrees. How high is the mountain in feet? 40. A carpenter designs a ramp for wheelchairs that forms an angle of 5 degrees with the floor. If the ramp must elevate the wheelchair a total of 2 feet, how long must he make the ramp? IX. You should be able to use the Fundamental Identities to simplify each expression. 4. Simplify: cos! + + cos! +

5 42. Simplify: cot! + csc! 43. Simplify: ( sec! " tan! )(sec! + tan!) 44. Simplify: 2sin 2! " cos 2! " " cos 2! 45. Simplify: ( + )( " ) X: You should be able to use cofunction identities to solve problems like the following: 46. cos45! = sin 47. tan 2 = cot 48. csc 63 = sec 49. cot! = tan 50. sec.2 = csc 5. sin (! /3) = cos 5