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Name of Lecturer: Mr. J.Agius Course: HVAC1 Lesson 52 Chapter 10: Trigonometry Sine, Cosine and Tangent Three Functions, but same idea. First Rule: Sine, Cosine and Tangent are all based on a Right-Angled Triangle Second Rule: Before getting stuck into the functions, it helps to give a name to each side of a right triangle: "Opposite" is opposite to the angle θ "Adjacent" is adjacent (next to) to the angle θ "Hypotenuse" is the long one Adjacent is always next to the angle And Opposite is opposite the angle 10 Trigonometry Page 1 Name of Lecturer: Mr. J.Agius Course: HVAC1 Sine, Cosine and Tangent The three main functions in trigonometry are Sine, Cosine and Tangent. They are often shortened to sin, cos and tan. To calculate them: Divide the length of one side by another side ... but you must know which sides! For a triangle with an angle θ, the functions are calculated this way: Sine Function: Cosine Function: Tangent Function: Example: What is the sine of 35°? Using this triangle (lengths are only to one decimal place): ( ) Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button. But you still need to remember what they mean! 10 Trigonometry Page 2 Name of Lecturer: Mr. J.Agius Course: HVAC1 Examples Special types of angles Example 1: what are the sine, cosine and tangent of 30°? The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √(3): Now we know the lengths, we can calculate the functions: ( Sine Cosine ( ) Tangent ( ) ) √ √ (get your calculator out and check them!) Example 2: what are the sine, cosine and tangent of 45° ? The classic 45° triangle has two sides of 1 and a hypotenuse of √(2): Sine ( ) Cosine ( ) Tangent 10 Trigonometry √ √ ( ) Page 3 Name of Lecturer: Mr. J.Agius Course: HVAC1 How can we remember these equations? Sohcahtoa Sohca...what? Just an easy way to remember which side to divide by which! Like this: Soh... ...cah... ...toa You can read more about sohcahtoa ... ... but please remember "sohcahtoa" - it could help in an exam ! Why? Why are these functions important? Because they let you work out angles when you know sides And they let you work out sides when you know angles Less Common Functions To complete the picture, there are 3 other functions where you divide one side by another, but they are not so commonly used. They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan: Secant Function: Cosecant Function: Cotangent Function: 10 Trigonometry Page 4 Name of Lecturer: Mr. J.Agius Course: HVAC1 Exercise 1 Use a calculator to solve. Round each answer to the nearest hundredth 1. cos 73 2. cos 74.5 3. tan 81 4. sin 59 5. sin 33 6. tan 75 7. tan 38.9 8. tan 57 9. cos 17 10. cos 4 11. sin 85 12. sin 23.5 13. sin 45 14. tan 9.8 15. sin 27 16. cos 67 17. tan 31 18. tan 63 19. cos 89.4 20. tan 49 21. sin 12.5 Exercise 2 Use a calculator to solve. Round to the nearest degree 1. tan-1 0.47 2. sin-1 0.99 3. cos-1 1 4. tan-1 3.31 5. cos-1 0.44 6. sin-1 0.47 7. cos-1 0.1 8. tan-1 0.53 9. tan-1 4.33 10. cos-1 0.4 11. sin-1 0.34 12. sin-1 0.31 13. sin-1 0.87 14. sin-1 1 15. tan-1 0.97 16. tan-1 5.67 17. cos-1 0.86 18. cos-1 0.81 19. sin-1 0.97 20. cos-1 0.02 21. sin-1 0.28 10 Trigonometry Page 5 Name of Lecturer: Mr. J.Agius Course: HVAC1 Exercise 3 Complete. Round the answer to the nearest hundredth 1. Find Cos U QG = 14.2 GU = 5.6 QU = 15.26 2. Find Tan B 3. Find Sin T BM = 3.6 VM = 5.1 BV = 6.24 AT = 0.41 AX = 0.28 TX = 0.3 5. Find Cos Y 6. Find Tan R UT = 8.62 UN = 8 TN = 3.2 YJ = 4.55 YR = 3.8 RJ = 2.5 PA = 0.76 AR = 2.5 PR = 2.61 7. Find Tan Y 8. Find Cos R 9. Find Sin X 4. Find Sin T VY = 20.94 VJ = 10.7 YJ = 18 10 Trigonometry KR = 27.5 SR = 29.26 SK = 10 KX = 4.12 KE = 0.97 XE = 4 Page 6 Name of Lecturer: Mr. J.Agius Course: HVAC1 Exercise 4 Find the value of x. 1. 2. N = 33.06 NY = x YC = 5.4 4. 3. C = 52.67 TY = 5.9 CY = x 5. V = 55.05 VJ = 5.8 LJ = x 10 Trigonometry E = 49.42 PU = 12.25 PE = x 6. A = 44.99 AE = 13.5 DA = x M = 50.39 MD = x MP = 0.51 Page 7 Name of Lecturer: Mr. J.Agius Course: HVAC1 Exercise 5 Complete. Round to the nearest hundredth 1. 2. BZ = 4 CB = 5.12 CZ = 3.2 JL = 10.48 JY = 8.3 YL = 6.4 ( ) 4. 3. ( ) 5. MA = 1.72 PM = 1.85 PA = 0.68 ( ) 10 Trigonometry MP = 3.76 DP = 4.33 MD = 5.68 ( ) 6. ZL = 7.36 ZK = 5.3 LK = 5.1 ( ) BK = 17.7 GK = 17.7 BG = 25.03 ( ) Page 8