Trigonometry Practice Problems with Answers

Answers: 1. cos α = 4/5 sin α = 3/5 tan α = 3/4 √ 2. sin ν = √11/6 tan ν = 11/5 cos ν = 5/6 3. sin ν = 1√ /2 tan ν = √ 3/3 cos ν = 3/2 4. sin β = b/c tan β = b/a cos β = a/c √ 5. sin α = √6/3 tan α = √ 2 cos α = 3/3 6. sin α = 4/5 tan α = 4/3 cos α = 3/5 √ 7. cot ν = 5√ 11/11 csc ν = 6 11/11 sec ν = 6/5 √ 8. cot ν = 3 csc ν = 2√ sec ν = 2 3/3 9. cot β = a/b csc β = c/b sec β = c/a √ 10. cot α = √ 2/2 sec α = √3 csc α = 6/2 11. cot α = 3/4 sec α = 5/3 csc α = 5/4 √ 12. sin α = 5 √34/34 cos α = 3 34/34 tan α = 5/3 cot α = √ 3/ 5 csc α = √34/5 sec α = 34/3 √ sin ω = 3 √34/34 cos ω = 5 34/34 tan ω = 3/5 cot ω = √ 5/ 3 csc ω = √34/3 sec ω = 34/5 √ 13. sin α = √3/3 cos α = √6/3 tan α = √ 2/2 cot α = √ 2 csc α = √3 sec α = 6/2 √ sin ω = √6/3 cos ω = √3/3 tan ω = √ 2 cot ω = √ 2/2 sec ω = √6/2 csc ω = 3 √ 14. sin α = √2/2 cos α = 2/2 tan α = 1 cot α = √ 1 csc α = √2 sec α = 2 √ 15. sin α = a√ 1 + a2 /(1 + a2 ) cos α = 1 + a2 /(1 + a2 ) tan α = a cot α = √ 1/a csc α = √1 + a2 /a sec α = 1 + a2 √ sin β = √ 1 + a2 /(1 + a2 ) cos β = a 1 + a2 /(1 + a2 ) tan β = 1/a cot β = √ a csc β = √1 + a2 sec β = 1 + a2 /a 16. csc β = 17/15 tan β = 15/8 cos α = 8/17 √ 17. sin θ = 2 √10/7 csc θ = 2√10/3 sec θ = 7 10/20 √ 18. sin ν = 2√ 5/5 cos ν = 5/5 cot ν = 1/2 19. sin α = 1√/2 cos α = √3/2 cot α = 3 √ 20. sin α = √1 − a2 tan α = 1 − a2 /a cos α = a 21. cos α = 4/5 sin2 α + cos2 α = 1 22. sin2 α + cos2 α = 1 23. sec2 α − tan2 α = 1 √ √ 24. x = 2 3/3, y = 4 3/3; 32. 1; 42. θ = 45◦ ; √ 25. x = 3/2, y = 3 3/3, ◦ α = 30 ; 33. 1; 43. θ = 30◦ ; 34. 1; 44. θ = 45◦ ; 35. 2; 45. θ = 30◦ ; 36. θ = 45◦ ; 46. θ = 60◦ ; 37. θ = 45◦ ; 47. θ = 60◦ ; 38. θ = 30◦ ; 48. θ = 30◦ ; 39. θ = 60◦ ; 49. θ = 60◦ ; 40. θ = 30◦ ; 50. θ = 45◦ ; 41. θ = 45◦ ; 51. θ = 30◦ ; √ 26. x = 4 2;, α = 45◦ ; √ 27. x = 6 3, y = 12;, α = 30◦ ; 28. √ 1+ 3 ; 2 29. 0; 30. 0; 31. √ √ 12−3 2+2 3 ; 6 Trigonometry 1. Find sin α, cos α and tan α in the following triangle: adjacent = hypotenuse opposite sin α = hypotenuse = opposite tan α = adjacent = 4. Find sin β, cos β and tan β in the following triangle: cos α = 15 9 α 12 cos β = tan β = c b β sin β = a 2. Find sin ν, cos ν and tan ν in the following triangle: 6 √ 11 ν sin ν = tan ν = cos ν = 5 3. Find sin ν, cos ν and tan ν in the following triangle: 5. Find sin α, cos α and tan α in the following triangle: √ 3 sin α = α √ tan α = 1 2 cos α = 6. Find sin α, cos α and tan α in the following triangle: sin α = tan α = ν √ 2 3 1 sin ν = tan ν = cos ν = cos α = 3 α 4 5 7. Referring to exercise #2, find the values of the other three trigonometric functions. 8. Referring to exercise #3, find the values of the other three trigonometric functions. 9. Referring to exercise #4, find the values of the other three trigonometric functions. 10. Referring to exercise #5, find the values of the other three trigonometric functions. 11. Referring to exercise #6, find the values of the other three trigonometric functions. Find the six trigonometric functions of each angle indicated in the following triangles: 12. 14. α ω a 5 a α 15. 3 √ 13. β 1 + a2 ω √ 1 3 α 1 α 16. Suppose sin β = 15/17. Find the values of csc β, tan β and cos β . 17. Suppose cos θ = 3/7. Find the values of sin θ, tan θ and csc θ. 18. Suppose tan ν = 12/6. Find the values of sin ν, cos ν and cot ν . 19. Suppose csc α = 2. Find the values of sin α, cos α and cot α. 20. Suppose sec α = 1/a. Find the values of sin α, tan α and cos α in terms of a. 21. Suppose sin α = 3/5. Find the value of cos α and the value of the expression sin2 α + cos2 α. 22. Referring to the picture below, how much is sin2 α + cos2 α? Completely simplify your answer. (Hint: c2 = a2 + b2 ). c α b a 23. Referring to the picture in exercise 22, how much is sec2 α − tan2 α? In the right-angled triangles below, find the values of the unknowns: 24. 26. 8 α x y x 30◦ 27. 2 25. x 6 60◦ 60◦ α x 3 α x y y Evaluate: 28. sin 30◦ + cos 30◦ ; 32. sin2 30◦ + cos2 30◦ (Hint: exercise #18); 29. sin 30◦ − cos 60◦ ; 33. tan 30◦ · cot 30◦ ; 30. sin2 45◦ + sin 30◦ − tan 45◦ ; 34. csc2 30◦ − cot2 30◦ ; 31. csc 30◦ − cos 45◦ + cot 60◦ ; 35.  tan2 60◦ − cot 45◦ 2 − sec2 45◦ ; Find the acute angle θ given: 1 36. sin θ = √ 2 √ 2 37. cos θ = 2 √ 3 38. cos θ = 2 40. cot θ = 39. sec θ = 2 43. tan θ = √ 3 41. cot θ = 1 42. tan θ = 1 √ 3 3 44. csc θ = √ 45. sec θ = √ 2 3 3 46. sin θ = 47. cos θ = 2 48. csc θ = 2 49. tan θ = √ 3 2 50. sec θ = √ 1 2 51. sin θ = 1 2 √ 3 2