Imp Ratios
Trigonometry ratios of these angles are important
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| Sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| Cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| Tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
| Cot θ | ∞ | √3 | 1 | 1/√3 | 0 |
| Sec θ | 1 | 2/√3 | √2 | 2 | ∞ |
| Cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |
Important relation b/w angles of right-angled triangle in terms of trigonometric ratio.
Graph indicating value of angle b/w which Trigonometric ratios are +ve
This concept becomes crucial as all the values of any trigonometric ratios lies b/w 0°and 90° only, and after that the same values are repeated which may be +ve or -ve. Like all the values of sinθ is b/w 0° to 90° only; after 90°, values are repeated though they may be +ve or -ve . From the graph, we can also analyze that after θ = 360°, values are repeated means value of sin30° = sin390° (360° +30°) = 1/2.
Concept for negative angle
Reduction of Trigonometric ratios of higher angle to smaller angle
Step 1: Divide higher angle in multiple of 90° and calulate value of θ as per the table below
| when n is odd number | when n is even number |
|---|---|
| sin(90n ± θ) = ± cos θ | sin(90n ± θ) = ± sin θ |
| cos(90n ± θ) = ± sin θ | cos(90n ± θ) = ± cos θ |
| tan(90n ± θ) = ± cot θ | tan(90n ± θ) = ± tan θ |
| cot(90n ± θ) = ± tan θ | cot(90n ± θ) = ± cot θ |
| sec(90n ± θ) = ± cosec θ | sec(90n ± θ) = ± sec θ |
| cosec(90n ± θ) = ± sec θ | cosec(90n ± θ) = ± cosec θ |
eg: sin 120° = sin (90° + 30°) = +cos30° (+ve as 120° falls in 2nd quadrant where sin is +ve)
tan 765° = tan (90° x 8 + 45°) = +tan 45° (+ve as 765° falls in 3rd quadrant where tan is +ve)
cosec 1410° = cosec (90° x 16 - 30°) = -cosec30 (-ve as 1410° falls in 4th quadrant where cosec is -ve)
tan 765° = tan (90° x 8 + 45°) = +tan 45° (+ve as 765° falls in 3rd quadrant where tan is +ve)
cosec 1410° = cosec (90° x 16 - 30°) = -cosec30 (-ve as 1410° falls in 4th quadrant where cosec is -ve)
Other Imp ratios
Apart from above angle, it is recommended that student should learn trigonometric ratios of these angles
N.Imp is not much important for competitive exams.
| Angle | 15° | 75° | 18° | 72° | 36° | 54° |
|---|---|---|---|---|---|---|
| Sin θ | (√3-1)⁄2√2 | (√3+1)⁄2√2 | (√5-1)⁄4 | N.Imp | N.Imp | (√5+1)⁄4 |
| Cos θ | (√3+1)⁄2√2 | (√3-1)⁄2√2 | N.Imp | (√5+1)⁄4 | (√5-1)⁄4 | N.Imp |
| Tan θ | 2-√3 | 2+√3 | N.Imp | N.Imp | N.Imp | N.Imp |
| Cot θ | 2+√3 | 2-√3 | N.Imp | N.Imp | N.Imp | N.Imp |
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