Match each trig function with its correct value if θ is an acute angle and cscθ = 2 1/2. match the following: square root of 21/5, 2/5, square root of 21/2, 2 and square root of 21/21, 5 and square root of 21/21 with secθ, cosθ, sinθ, cotθ, tanθ

To solve this problem, we are given that cscθ = 2 1/2.

1. First, let's find the value of sinθ. Since cscθ is the reciprocal of sinθ, we can find sinθ by taking the reciprocal of 2 1/2.

Reciprocal of 2 1/2 = 1 / (2 1/2) = 1 / (5/2) = 2/5

So, sinθ = 2/5.

2. Next, let's find the value of cosθ. We know that sinθ = opposite/hypotenuse and cosθ = adjacent/hypotenuse. Since θ is an acute angle, we can use the Pythagorean theorem to find the value of cosθ.

Using the Pythagorean theorem, we have: sin^2θ + cos^2θ = 1

Substituting the value of sinθ, we get: (2/5)^2 + cos^2θ = 1 4/25 + cos^2θ = 1 cos^2θ = 1 - 4/25 cos^2θ = 21/25

Taking the square root of both sides, we get: cosθ = ± √(21/25)

Since θ is an acute angle, cosθ will be positive. Therefore, cosθ = √(21/25) = √21/5.

3. Now, let's find the value of tanθ. We know that tanθ = sinθ/cosθ. Substituting the values we found earlier, we get: tanθ = (2/5) / (√21/5) tanθ = 2/√21

4. Next, let's find the value of cotθ. We know that cotθ = 1/tanθ. Substituting the value of tanθ, we get: cotθ = 1 / (2/√21) cotθ = √21/2

5. Finally, let's find the value of secθ. We know that secθ = 1/cosθ. Substituting the value of cosθ, we get: secθ = 1 / (√21/5) secθ = 5/√21

So, matching the trig functions with their values, we have: sinθ = 2/5 cosθ = √21/5 tanθ = 2/√21 cotθ = √21/2 secθ = 5/√21