Question

GIVEN THAT A IS ACUTE AND 2 SIN² A - 1/3 = COS ² A .FIND ANGEL A

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the value of angle A given the equation 2sin²A - 1/3 = cos²A, we can use trigonometric identities to simplify the equation.

We know that sin²A + cos²A = 1 (Pythagorean identity) and cos²A = 1 - sin²A.

Substitute the value of cos²A in the equation:

2sin²A - 1/3 = 1 - sin²A

Now, simplify the equation:

3sin²A - 1 = 3 - 3sin²A
6sin²A = 4
sin²A = 4/6
sin²A = 2/3

Take the square root of both sides to find sinA:

sinA = √(2/3)

Now, find the value of angle A by taking the inverse sine function (sin⁻¹) of √(2/3):

A = sin⁻¹(√(2/3))

Therefore, the value of angle A in radians is sin⁻¹(√(2/3)).

Answer 2

somehow you went off the rails. Let's start with

2sin²A - 1/3 = 1 - sin²A
3sin²A = 4/3
sin²A = 4/9
sinA = ±2/3
if sinA = 2/3 then A ≈ 41.8° or 138.2°
if sinA = -2/3 then A ≈ 221.8° or 318.2°

Answer 3

[AI-generated answer — Bot GPT 3.5]

Apologies for the mistake in my previous response. Thank you for correcting it.

Given the equation 2sin²A - 1/3 = 1 - sin²A, following your steps, we arrive at:

3sin²A = 4/3
sin²A = 4/9
sinA = ±2/3

If sinA = 2/3, then Angle A can be approximately 41.8° or 138.2°.

If sinA = -2/3, then Angle A can be approximately 221.8° or 318.2°.

Thank you for pointing out the error.