How do you find sin(2x), cos(2x), and tan(2x) when given that sin(x) = 0.35 and x is in the first quadrant?

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To find sin(2x), cos(2x), and tan(2x) given that sin(x) = 0.35 and x is in the first quadrant, we can use some trigonometric identities and properties.

Step 1: Find cos(x)

Since we know sin(x), we can find cos(x) using the Pythagorean identity:

sin²(x) + cos²(x) = 1

Substituting the value of sin(x):

0.35² + cos²(x) = 1

Calculating:

0.1225 + cos²(x) = 1

Thus:

cos²(x) = 1 - 0.1225 = 0.8775

Now take the square root to find cos(x):

cos(x) = sqrt(0.8775) ≈ 0.935

Step 2: Find sin(2x) and cos(2x)

We can use the double angle formulas for sine and cosine:

sin(2x) = 2 * sin(x) * cos(x)
cos(2x) = cos²(x) - sin²(x)

Substituting the values we have:

sin(2x) = 2 * 0.35 * 0.935 ≈ 0.655
cos(2x) = 0.935² - 0.35² ≈ 0.875

Step 3: Find tan(2x)

The tangent of the double angle can be calculated using:

tan(2x) = sin(2x) / cos(2x)

Substituting the values:

tan(2x) = 0.655 / 0.875 ≈ 0.749

Final Answers

  • sin(2x) ≈ 0.655
  • cos(2x) ≈ 0.875
  • tan(2x) ≈ 0.749

In summary, using the given information and trigonometric identities, we successfully computed sin(2x), cos(2x), and tan(2x).

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