Find the remaining trigonometric ratios. tan(α) = 2, 0<α<π/2the responses they want are:1. sine(α)=_2. cos(α)=_3. cot(α)=_4. sec(α)=_5. csc(α)=_

1. sinα = +2√5/5 2.  cosα = +√5/5 3. cotα = +1/2 4. secα = +√5 5. cosecα = +√5/2.

Step-by-step explanation:

1. Since tan(α) = 2 and 1 + cot²α = cosec²α

1 + 1/tan²α = 1/sin²α

1 + 1/2² = 1/sin²α

1 + 1/4 = 1/sin²α

5/4 = 1/sin²α

sinα = ±√(4/5)

sinα = ±2/√5

sinα = ±2√5/5

Since 0<α<π/2, sinα = +2√5/5

2. sin²α + cos²α = 1

(2/√5)² + cos²α = 1

4/5 + cos²α = 1

cos²α = 1 - 4/5

cos²α = 1/5

cosα = ±1/√5

cosα = ±√5/5

Since 0<α<π/2, cosα = +√5/5

3. cotα = 1/tanα = 1/2

Since 0<α<π/2, cotα = +1/2

4. secα = 1/cosα = 1/±1/√5 = ±√5

Since 0<α<π/2, secα = +√5

5. cosecα = 1/sinα = 1/±2/√5 = ±√5/2

Since 0<α<π/2, cosecα = +√5/2

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