SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios Special Right Triangles Right Triangles: MATH 1330 Precalculus 345
CHAPTER 4 Trigonometric Functions 45 o -45 o -90 o Triangles: Theorem for 45 o -45 o -90 o Triangles: Eample: 346 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios 30 o -60 o -90 o Triangles: Theorem for 30 o -60 o -90 o Triangles: Eample: MATH 1330 Precalculus 347
CHAPTER 4 Trigonometric Functions Additional Eample 1: Part (a): Part (b): 348 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Eample 2: MATH 1330 Precalculus 349
CHAPTER 4 Trigonometric Functions Additional Eample 3: Part (a): Part (b): 350 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Eample 4: MATH 1330 Precalculus 351
CHAPTER 4 Trigonometric Functions Additional Eample 5: 352 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Trigonometric Ratios The Three Basic Trigonometric Ratios: Eample: MATH 1330 Precalculus 353
CHAPTER 4 Trigonometric Functions 354 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios The Three Reciprocal Trigonometric Ratios: MATH 1330 Precalculus 355
CHAPTER 4 Trigonometric Functions Eample: 356 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Eample 1: Part (a): Part (b): MATH 1330 Precalculus 357
CHAPTER 4 Trigonometric Functions Part (c): Additional Eample 2: 358 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios MATH 1330 Precalculus 359
CHAPTER 4 Trigonometric Functions Additional Eample 3: 360 University of Houston Department of Mathematics
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Eample 4: MATH 1330 Precalculus 361
CHAPTER 4 Trigonometric Functions 362 University of Houston Department of Mathematics
Eercise Set 4.1: Special Right Triangles and Trigonometric Ratios Answer the following. 9. 1. If two sides of a triangle are congruent, then the opposite those sides are also congruent. 4 2 45 o 2. If two angles of a triangle are congruent, then the opposite those angles are also congruent. 3. In any triangle, the sum of the measures of its angles is degrees. 10. 45 o 3 2 4. In an isosceles right triangle, each acute angle measures degrees. 5. Fill in each missing blank with one of the following: smallest, largest In any triangle, the longest side is opposite the angle, and the shortest side is opposite the angle. 6. Fill in each missing blank with one of the following: 30 o, 60 o, 90 o In a 30 o -60 o -90 o triangle, the hypotenuse is opposite the angle, the shorter leg is opposite the angle, and the longer leg is opposite the angle. 11. 12. 45 o 45 o 8 7 For each of the following, (a) Use the theorem for 45 o -45 o -90 o triangles to find. (b) Use the Pythagorean Theorem to verify the result obtained in part (a). 13. 9 7. 5 45 o 14. 12 8. 45 o 15. 8 8 2 MATH 1330 Precalculus 363
Eercise Set 4.1: Special Right Triangles and Trigonometric Ratios 16. 45 o 2 3 22. In the figure below, an altitude is drawn to the base of an equilateral triangle. (a) Find a and b. (b) Justify the answer obtained in part (a). (c) Use the Pythagorean Theorem to find c, the length of the altitude. (Write c in simplest radical form.) 17. 2 3 45 o 60 o a 30 o 30 o c 60 o b 4 18. For each of the following, Use the theorem for 30 o -60 o - 90 o triangles to find and y. 5 2 23. 30 o y The following eamples help to illustrate the theorem regarding 30 o -60 o -90 o triangles. 7 19. What is the measure of each angle of an equilateral triangle? 20. An altitude is drawn to the base of the equilateral triangle drawn below. Find the measures of and y. 24. 30 o 22 y y o 25. o 6 3 21. In the figure below, an altitude is drawn to the base of an equilateral triangle. (a) Find a and b. (b) Justify the answer obtained in part (a). (c) Use the Pythagorean Theorem to find c, the length of the altitude. 26. y y 60 o 30 o 60 o 30 o 30 o c 60 o 10 8 a b 364 University of Houston Department of Mathematics
Eercise Set 4.1: Special Right Triangles and Trigonometric Ratios 27. 60 o y 5 (b) Find the following: sin cos tan A sin A cos A tan B B B 28. y 15 3 60 o 34. D 25 29. y 30 o 6 E (a) Use the Pythagorean Theorem to find DE. (b) Find the following: sin cos tan 24 F D sin D cos D tan F F F 30. 31. 60 o y 4 2 y 60 o 5 3 35. Suppose that is an acute angle of a right 5 triangle and sin. Find cos and 7 tan. 36. Suppose that is an acute angle of a right triangle and tan cos. 4 2 7. Find sin and 32. 8 y 30 o 37. The reciprocal of the sine function is the function. 38. The reciprocal of the cosine function is the function. Answer the following. Write answers in simplest form. 39. The reciprocal of the tangent function is the function. 33. A 15 17 40. The reciprocal of the cosecant function is the function. 41. The reciprocal of the secant function is the function. C B (a) Use the Pythagorean Theorem to find BC. 42. The reciprocal of the cotangent function is the function. MATH 1330 Precalculus 365
Eercise Set 4.1: Special Right Triangles and Trigonometric Ratios 43. 6 5 (a) Use the Pythagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of. 47. Suppose that is an acute angle of a right 2 10 triangle and cot. Find the si 3 trigonometric functions of. 48. Suppose that is an acute angle of a right 5 triangle and sec. Find the si 2 trigonometric functions of. 44. 45. 46. 7 4 (a) Use the Pythagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of. 4 8 (a) Use the Pythagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of. 8 6 (a) Use the Pythagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of. 49. (a) Use the theorems for special right triangles to find the missing side lengths in the triangles above. (b) Using the triangles above, find the following: sin 45 cos 45 tan 45 csc45 sec45 cot 45 (c) Using the triangles above, find the following: sin 30 cos 30 tan 30 csc30 sec30 cot 30 (d) Using the triangles above, find the following: sin 60 cos 60 3 tan 60 45 o 60 o 30 o 2 csc60 sec60 cot 60 366 University of Houston Department of Mathematics
Eercise Set 4.1: Special Right Triangles and Trigonometric Ratios 50. 8 45 o 12 30 o 60 o (a) Use the theorems for special right triangles to find the missing side lengths in the triangles above. (b) Using the triangles above, find the following: sin 45 cos 45 tan 45 csc45 sec45 cot 45 (c) Using the triangles above, find the following: sin 30 cos 30 tan 30 csc30 sec30 cot 30 (d) Using the triangles above, find the following: sin 60 cos 60 tan 60 csc60 sec60 cot 60 51. Compare the answers to parts (b), (c), and (d) in the previous two eamples. What do you notice? MATH 1330 Precalculus 367