Trigonometrical Identities
1. Quotient relations:
- tan = sin /cos
- cot = cos /sin
2. Reciprocal relations:
- sin = 1/ cosec
- cosec = 1/sin
- cos = 1/sec
- sec = 1/cos
- tan = 1/cot
- cot = 1/tan
3. Square relations (Fundamental Identities):
- sin² +cos² = 1
- 1 +tan² = sec²
- 1 +cot² = cosec²
4. T-ratios of standard angles:
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0° |
30° |
45° |
60° |
90° |
Sin |
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Cos |
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5. T-ratios of complementary angles:
- sin (90° - ) = cos
- cos (90° - ) = sin
- tan (90° - ) = cot
- cot (90° - ) = tan
- sec (90° - ) = cosec
- cosec (90° - ) = sec
Exercise
Prove the following identities (1 -15):
- tan² - 1/cos² +1 = 0
- sin A/(1 +cosA) +(1 +cos A)/sin A = 2 cosecA
- (1 -sin )/(1 +sin )
= (sec
-tan )²
- sec² A +cosec² A = sec² A cosec²A
- sec4 -tan4
= 1 +2 tan²
- tan² A/(1+tan² A) + cot² A/(1 +cot² A) = 1
- (1 -cos )(1 +sec )
= tan sin
- (cot² A -tan² A)/(cot A +tan A)² = 2 cos² A -1
- cos4 +sin4
+2 sin² cos² = 1
-
= cosec +cot
- .
- sec (1 -sin )(sec
+tan ) =1
- (sec A +cos A)(sec A -cos A) = tan²A +sin² A
- tan² -sin² = tan²
sin²
- (1 -cos )(1 + cos )(1 +cot²
) = 1
Simplify the expression in questions (16 -27):
- (sin² A -cos² A)/(sin A -cos A)
- sin² cos cosec³
sec
- cot B sin² B cot B
- (cos² A +cos A -12)/cos A - 3)
- tan /(sec -1)
+ tan /(sec +1)
- sec A csc A -tan A -cot A
- (cot² +tan² )/(cos²
sec² )
- x = a sec , y = b tan
- x = h +a cos , y = k
+b sin
- x = a sec³ , y = b tan³
- tan +sin = m, tan
- sin = n
- cot +cos = m, cot
-cos = n
When 0° < < 90°, solve the following equations (28-33):
- 2 sin² = 1/2
- 4 cos² -3 = 0
- sin² -(1/2)sin = 0
- tan² = 3 (sec -1)
- 2 cosec = 3 sec²
- 3 tan +cot = 5 cosec
Without using trigonometric tables, evaluate (34-43):
- (cos0° +sin45° +sin30°).(sin90° -cos45° +cos60°)
- sin² 45° -tan² 60° + cos² 90°
- sin 23°/cos 67°
- cosec 31°/ sec 59°
- sin² 38° -sin²52°
- sin 18° -cos 72°
- sin 36° sec 54° +cos 24° cosec 66°
-
.
- cosec²67° -tan² 23°
- sec 31° sin 59° +cos 31° cosec 59°
Express the following in terms of t-ratios of angles between 0° and 45°.
- sin 85° +cosec 85°
- cosec 69° +cot 69°
- sin 81° +tan 81°
- cos 56° +cot 56°
Prove the following:
- [sin (90 -A) sin A]/tan A-1 = - sin² A
- cos cos(90° - ) -sin
sin (90° - ) = 0
- sin (90° - ) cos (90° - )
= tan /(1 +tan² )
- cosec² (90° - ) -tan²
= cos²(90° - ) +cot²
- If cos/cos = m and
cos/sin = n, show that (m²
+n²) cos² = n².
- If x = r cos sin,
y = r cos cos and z
= r sin , show that x² +y² +z²
= r².
Answers
16. sin A +cos A
17. csc
18. cos² B
19. cos A +4
20. 2 csc
21. 0
22. 1
23. x²/a² - y²/b² = 1
24.
25.
26. (m² -n²)² = 16 mn
27. (m² -n²)² = 16 mn
28. 30°
29. 30°
30. 30°
31. 60°
32. 30°
33. 60°
34. -5/2
35. 1
36. 1
37. 1
38. 0
39. 2
40. 1/2
41. 1
42. 2
43. 0
44. sec 21° +tan 21°
45. cos 9° +cot 9°
46. sin 34° +tan 34°
47. sin²