Trigonometry Solved Problems

Trigonometry is the branch of mathematics dealing with the relationship of sides and angles of triangles

Subjects > Mathematics > Trigonometry

Which problem do you want to solve ?

  1. Show that `tan theta + cot theta = sec theta cosec theta`
  2. Prove that: `tan^2 A - sin^2 A = sin^4 A sec^2 A`
  3. Prove that: `sin 330^0 cos 390^0 - cos 570^0 sin 510^0 = 0`
  4. Show that: `sin (270^0 - theta) - sin (270^0 + theta) = cos theta + cos (180^0 + theta)`
  5. If `sin theta = (a^2 - b^2)/(a^2 + b^2)` find the values of `cos theta` and `tan theta`
  6. Find the value of:
    `sin^2 A cosec ((pi)/2 - A) - cot^2 ((pi)/2 - A)cos A`
  7. If `theta` is an angle in the first quadrant and `tan theta = t`, express all other trig ratios in terms of `t`.
  8. Prove that:
    `(1 + sec x + tan x)(1 + co s e c x + cot x) = 2(1 + tan x + cot x + sec x + c o s e c x)`
  9. If `tan^2 alpha - 2tan^2 beta = 1`, find the possible values of `cos alpha to cos beta`
  10. If `sec theta - cos theta = a` and `cosec theta - sin theta = b`, prove that: `a^2 b^2 (a^2 + b^2 + 3) = 1`
  11. Show that:
    `(cos theta - 1)/(sec theta + tan theta) + (cos theta + 1)/(sec theta - tan theta) = 2(1 + tan theta)`
  12. Prove that: `(cot alpha + tan beta)/(cot beta + tan alpha) = cot alpha tan beta`
  13. Prove that:
    `(1 + sin theta)/(cos theta) = (cos theta)/(1 - sin theta) = sec theta + tan theta`
  14. If `xcos theta + ysin theta = a` and `xsin theta - ycos theta = b`, prove that:
    `tan theta = (bx + ay)/(ax - by)` and `x^2 + y^2 = a^2 + b^2`
  15. If `tan theta + sin theta = x` and `tan theta - sin theta = y`, prove that `(x^2 - y^2)^2 = 16xy`
  16. Show that: `cos (alpha + beta) cos (alpha - beta) = cos^2 alpha sin^2 beta`
  17. Prove that: `cot (A + B) = (cot Acot B - 1)/(cot A + cot B)`
  18. Show that: `sin (A + B + C) = cos Acos Bcos C (tan A + tan B + tan C - tan Atan Btan C)` and deduce that if `A`, `B`, and `C` are the angles of triangle then `cot Acot B + cot Bcot C + cot Ccot A = 1`
  19. Express: `1 + sin 2x` in factors.
  20. Express: `sin A + cos B` in factors.
  21. Express: `cos A - sin B` in factors.
  22. Prove the following identity:
    `(cos B + cos C)/(sin B - sin C) = cot ((B - C)/2)`
  23. Prove that: `(cos B - cos C)/(sin B + sin C) = -tan ((B - C)/2)`
  24. Prove that: `(sin B + sin C)/(cos B + cos C) = tan ((B + C)/2)`
  25. Prove that:
    `(sin B - sin C)/(sin B + sin C) = cot ((B + C)/2) tan ((B - C)/2)`
  26. Prove that: `sin x + sin 2x + sin 3x = (2cos x + 1)sin 2x`
  27. Prove that: `cos x + sin 2x - cos 3x = (2sin x + 1)sin 2x`
  28. Prove that: `cos 3theta + cos 5theta + cos 7theta = cos 5theta * (2cos 2theta + 1)`
  29. Prove that: `1 + 2cos 2theta + cos 4theta = 4cos^2 theta cos 2theta`
  30. Prove that:
    `2tan^(-1) (1/8) + tan^(-1) ((11)/7) + 2tan^(-1) (1/5) = (pi)/4`
  31. Solve the for `x` in the range `0^0 <= x <= 290^0` given that: `cos 3x = sin 2x`
  32. Find the general solution of the equation: `tan 3theta = cot 2theta`
  33. Find all the angles between `0^0` and `360^0` which satisfy the equation: `6sin^2 x + 5cos x = 7`
  34. Find the general values of `x` satisfying the equation: `4cos x + 5 = 6sin^x`
  35. Assuming `r` is positive, find `r` and a value of `theta` between `-180^0` and `180^0` to satisfy the equation: `rcos theta = -4`, `rsin theta = 2.5`
  36. Find all the angles between `0^0` and `360^0` which satisfy the equation: `3tan^3 theta - 3tan^2 theta = tan theta - 1`
  37. Find the general solution of the equation: `(2tan x - 1)^2 = 3(sec^2 -2)`
  38. Find the general solution of the equation: `tan xtan 4x = 1`
  39. Find the values of `x`, in radians between `0` and `2pi` which satisfy the equation: `6tan^2 x - 4sin^2 x = 1`
  40. Find the values of `x` from `0^0` and `360^0` satisfying the equation `10sin^2 x + 10sin xcos x - cos^2 x = 2`
  41. Evaluate: `tan [sin^(-1) (3/4)]`
  42. Evaluate : `cos (sin^(-1) x)`
  43. If `8sin^2 theta + 2cos theta - 5 = 0` show that `cos theta = 3/4` or `(-1)/2`. Hence find the possible values of `tan theta`
  44. Solve for `theta`, when `-180^0 <= theta <= 180^0`, `cos theta + sin theta + 2 = 0`
  45. Prove that: `sqrt ((1 - sin theta)/(1 + sin theta)) = sec theta - tan theta`
  46. Express `sin 3theta` in terms of `sin theta`. Hence solve the equation: `sin^2 theta + sin theta = 1 - sin 3theta`
  47. Given that: `6cos^2 theta - 8sin theta cos theta = A + Rcos (2theta + alpha)` , find the values of the constants `A, R` and `alpha`
  48. Calculate the possible values of `tan theta` if:
    `cos theta = (20cos^4 theta - 24sin^2 theta + 6)/(10sin^3 theta - 7sin theta)`

Recommended Lessons

Which problem do you want to solve ?

  1. Show that `tan theta + cot theta = sec theta cosec theta`
  2. Prove that: `tan^2 A - sin^2 A = sin^4 A sec^2 A`
  3. Prove that: `sin 330^0 cos 390^0 - cos 570^0 sin 510^0 = 0`
  4. Show that: `sin (270^0 - theta) - sin (270^0 + theta) = cos theta + cos (180^0 + theta)`
  5. If `sin theta = (a^2 - b^2)/(a^2 + b^2)` find the values of `cos theta` and `tan theta`
  6. Find the value of:
    `sin^2 A cosec ((pi)/2 - A) - cot^2 ((pi)/2 - A)cos A`
  7. If `theta` is an angle in the first quadrant and `tan theta = t`, express all other trig ratios in terms of `t`.
  8. Prove that:
    `(1 + sec x + tan x)(1 + co s e c x + cot x) = 2(1 + tan x + cot x + sec x + c o s e c x)`
  9. If `tan^2 alpha - 2tan^2 beta = 1`, find the possible values of `cos alpha to cos beta`
  10. If `sec theta - cos theta = a` and `cosec theta - sin theta = b`, prove that: `a^2 b^2 (a^2 + b^2 + 3) = 1`
  11. Show that:
    `(cos theta - 1)/(sec theta + tan theta) + (cos theta + 1)/(sec theta - tan theta) = 2(1 + tan theta)`
  12. Prove that: `(cot alpha + tan beta)/(cot beta + tan alpha) = cot alpha tan beta`
  13. Prove that:
    `(1 + sin theta)/(cos theta) = (cos theta)/(1 - sin theta) = sec theta + tan theta`
  14. If `xcos theta + ysin theta = a` and `xsin theta - ycos theta = b`, prove that:
    `tan theta = (bx + ay)/(ax - by)` and `x^2 + y^2 = a^2 + b^2`
  15. If `tan theta + sin theta = x` and `tan theta - sin theta = y`, prove that `(x^2 - y^2)^2 = 16xy`
  16. Show that: `cos (alpha + beta) cos (alpha - beta) = cos^2 alpha sin^2 beta`
  17. Prove that: `cot (A + B) = (cot Acot B - 1)/(cot A + cot B)`
  18. Show that: `sin (A + B + C) = cos Acos Bcos C (tan A + tan B + tan C - tan Atan Btan C)` and deduce that if `A`, `B`, and `C` are the angles of triangle then `cot Acot B + cot Bcot C + cot Ccot A = 1`
  19. Express: `1 + sin 2x` in factors.
  20. Express: `sin A + cos B` in factors.
  21. Express: `cos A - sin B` in factors.
  22. Prove the following identity:
    `(cos B + cos C)/(sin B - sin C) = cot ((B - C)/2)`
  23. Prove that: `(cos B - cos C)/(sin B + sin C) = -tan ((B - C)/2)`
  24. Prove that: `(sin B + sin C)/(cos B + cos C) = tan ((B + C)/2)`
  25. Prove that:
    `(sin B - sin C)/(sin B + sin C) = cot ((B + C)/2) tan ((B - C)/2)`
  26. Prove that: `sin x + sin 2x + sin 3x = (2cos x + 1)sin 2x`
  27. Prove that: `cos x + sin 2x - cos 3x = (2sin x + 1)sin 2x`
  28. Prove that: `cos 3theta + cos 5theta + cos 7theta = cos 5theta * (2cos 2theta + 1)`
  29. Prove that: `1 + 2cos 2theta + cos 4theta = 4cos^2 theta cos 2theta`
  30. Prove that:
    `2tan^(-1) (1/8) + tan^(-1) ((11)/7) + 2tan^(-1) (1/5) = (pi)/4`
  31. Solve the for `x` in the range `0^0 <= x <= 290^0` given that: `cos 3x = sin 2x`
  32. Find the general solution of the equation: `tan 3theta = cot 2theta`
  33. Find all the angles between `0^0` and `360^0` which satisfy the equation: `6sin^2 x + 5cos x = 7`
  34. Find the general values of `x` satisfying the equation: `4cos x + 5 = 6sin^x`
  35. Assuming `r` is positive, find `r` and a value of `theta` between `-180^0` and `180^0` to satisfy the equation: `rcos theta = -4`, `rsin theta = 2.5`
  36. Find all the angles between `0^0` and `360^0` which satisfy the equation: `3tan^3 theta - 3tan^2 theta = tan theta - 1`
  37. Find the general solution of the equation: `(2tan x - 1)^2 = 3(sec^2 -2)`
  38. Find the general solution of the equation: `tan xtan 4x = 1`
  39. Find the values of `x`, in radians between `0` and `2pi` which satisfy the equation: `6tan^2 x - 4sin^2 x = 1`
  40. Find the values of `x` from `0^0` and `360^0` satisfying the equation `10sin^2 x + 10sin xcos x - cos^2 x = 2`
  41. Evaluate: `tan [sin^(-1) (3/4)]`
  42. Evaluate : `cos (sin^(-1) x)`
  43. If `8sin^2 theta + 2cos theta - 5 = 0` show that `cos theta = 3/4` or `(-1)/2`. Hence find the possible values of `tan theta`
  44. Solve for `theta`, when `-180^0 <= theta <= 180^0`, `cos theta + sin theta + 2 = 0`
  45. Prove that: `sqrt ((1 - sin theta)/(1 + sin theta)) = sec theta - tan theta`
  46. Express `sin 3theta` in terms of `sin theta`. Hence solve the equation: `sin^2 theta + sin theta = 1 - sin 3theta`
  47. Given that: `6cos^2 theta - 8sin theta cos theta = A + Rcos (2theta + alpha)` , find the values of the constants `A, R` and `alpha`
  48. Calculate the possible values of `tan theta` if:
    `cos theta = (20cos^4 theta - 24sin^2 theta + 6)/(10sin^3 theta - 7sin theta)`

Recommended Lessons