If cos A=1/3 and tan A<0,determine the value of 2sin A+tan A without the use of a calculator
we are in QIV, so sinA = -√8/3
Now just plug in the values for sinA and cosA to evaluate your expression.
To determine the value of 2sin A + tan A without a calculator, we need to find the values of sin A and tan A first using the given information.
Given that cos A = 1/3, we can use the Pythagorean identity sin^2 A + cos^2 A = 1 to find sin A. Rearranging the equation:
sin^2 A = 1 - cos^2 A
sin^2 A = 1 - (1/3)^2
sin^2 A = 1 - 1/9
sin^2 A = 8/9
Since sin A > 0 (sine is positive in the first and second quadrants), we take the positive square root:
sin A = √(8/9) = (√8)/3
Next, we need to determine the sign of tan A based on the given information tan A < 0. Since tan A < 0, we know that the tangent function is negative in either the second or fourth quadrant.
Now we can calculate 2sin A + tan A:
2sin A + tan A = 2 * √(8/9) + tan A
To determine whether to add or subtract √(8/9), we need to consider the sign of tan A. Since tan A < 0, we know that subtracting √(8/9) will give us the correct value for 2sin A + tan A. Therefore:
2sin A + tan A = 2 * (√8)/3 - √(8/9)
To simplify this expression further, we can rationalize the denominator of the second term:
2sin A + tan A = 2 * (√8)/3 - √(8/9) * (√9/√9)
= 2 * (√8)/3 - (√8)/3
= (√8 - √8)/3
= 0/3
= 0
Therefore, the value of 2sin A + tan A is 0.