Question

Find sin2x, cos2x, and tan2x if sinx = - 1/ √5 square root 10 and x terminates in quadrant III

Answer 1

Using double angle formulas and the Pythagorean identity, sin2x is calculated as 3/5, cos2x as 4/5, and tan2x as -3/4, where x is an angle in the third quadrant with given sinx = -1/√10.

Step-by-step explanation:

To find sin2x, cos2x, and tan2x given that sinx = -1/√10 and x terminates in quadrant III, we will use the double angle formulas:

• sin2x = 2sinx cosx
• cos2x = cos²x - sin²x = 1 - 2sin²x
• tan2x = ​​2tanx / (1 - tan²x)

Since sinx is given, we can find cosx using the Pythagorean identity cos²x = 1 - sin²x. In quadrant III, both sine and cosine are negative, so cosx = -√(1 - sin²x).

Therefore,
cosx = -√(1 - (-1/√10)²) = -√(1 - 1/10) = -√(9/10) = -3/√10

Now, we can use the double angle formulas:

• sin2x = 2(-1/√10)(-3/√10) = 6/10 = 3/5
• cos2x = 1 - 2(-1/√10)² = 1 - 2/10 = 8/10 = 4/5
• tan2x = 2tanx / (1 - tan²x) = 2(-1/3) / (1 - (-1/3)²) = -2/3 / (1 - 1/9) = -2/3 / (8/9) = -3/4

Answer 2

To find sin2x, cos2x, and tan2x when sinx is -1/√10 in quadrant III, use the double-angle formulas for sine and cosine, remembering the signs of functions in quadrant III. Calculations show sin2x = 1.8, cos2x = 0.8, and tan2x = 2.25.

Step-by-step explanation:

To find sin2x, cos2x, and tan2x given that sinx = -1/√10 and the angle x terminates in quadrant III, we use the double-angle formulas for sine and cosine, and then find the tangent from these results.

Double-Angle Formulas:

1. sin 2x = 2 sin x cos x
2. cos 2x can be computed using any of these:
3. tan 2x = sin 2x / cos 2x

Calculation for Quadrant III:

In quadrant III, both sine and cosine are negative. Thus, if sinx = -1/√10, we find cosx using the Pythagorean identity sin²x + cos²x = 1, which implies cos²x = 1 - sin²x. Here, cos²x = 1 - (-1/√10)² = 1 - 1/10 = 9/10, and since cosx is negative in quadrant III, cosx = -√(9/10).

Now apply the double-angle formulas:

sin 2x = 2 * (-1/√10) * (-√(9/10)) = 2/√10 * √(9/10) = 18/10 or sin 2x = 1.8

cos 2x = 1 - 2*(-1√10)² = 1 - 2/10 = 8/10 or cos 2x = 0.8

tan 2x = sin 2x / cos 2x = 1.8 / 0.8 = 2.25