Question

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)=_

Answer 1

To find the remaining trigonometric ratios, we need to use the value of tan(α) and the fact that 0α<π/2. By using the given information, we find sine(α), cosine(α), cotangent(α), secant(α), and cosecant(α) to be 2/√5, 1/√5, 1/2, √5, and √5/2 respectively.

Step-by-step explanation:

To find the remaining trigonometric ratios based on the given information, we need to use the value of tan(α) and the fact that 0α<π/2.

1. Use the Pythagorean identity: sin2(α) + cos2(α) = 1. Given that tan(α) = 2, we can use:

sin2(α) + (2/√(4+1))2 = 1. Solving this equation gives us sin(α) = 2/√5.

2. To find cos(α), we can use the equation: cos(α) = 1/√(1+tan2(α)). Substituting the value of tan(α) = 2, we get cos(α) = 1/√5.

3. We can find cot(α) by taking the reciprocal of tan(α): cot(α) = 1/tan(α). Substituting the given value of tan(α) = 2, we find cot(α) = 1/2.

4. Similarly, we can find sec(α) by taking the reciprocal of cos(α): sec(α) = 1/cos(α). Substituting the given value of cos(α) = 1/√5, we find sec(α) = √5.

5. Lastly, we can find csc(α) by taking the reciprocal of sin(α): csc(α) = 1/sin(α). Substituting the given value of sin(α) = 2/√5, we find csc(α) = √5/2.

Answer 2

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