Solve 4cos^2A = 3cosA for 90º≤A≤180º. (Enter only the number.)
Find A when 0º≤A≤90º and 3tan^2A = 2tanA + 1. (Enter only the number.)
let z = cos A
4 z^2 - 3 z = 0
z(4z-3) = 0
z = 0 or z = 3/4
so
cos A = 0 when z = 90 degrees
cos A = 3/4 when z = 41.4 degrees but that is not between 90 and 180
4cos^2 A - 3cosA = 0
cosA(4cosA - 3) = 0
cosA=0 or cosA = 3/4
but your domain is quadrant II, where the cosine is negative
except cos90° = 0
so x = 90°
do the 2nd the same way by factoring out tanA
To solve the equation 4cos^2A = 3cosA for 90º≤A≤180º, we can follow these steps:
Step 1: Rewrite the equation using the identity cos^2A = 1 - sin^2A.
4(1 - sin^2A) = 3cosA
Step 2: Substitute cosA with sqrt(1 - sin^2A) from the Pythagorean identity.
4(1 - sin^2A) = 3sqrt(1 - sin^2A)
Step 3: Simplify the equation by distributing the terms.
4 - 4sin^2A = 3sqrt(1 - sin^2A)
Step 4: Rearrange the equation, moving all terms to one side.
4sin^2A + 3sqrt(1 - sin^2A) - 4 = 0
Step 5: Let's solve this equation numerically by using a calculator or a software tool.
By solving the equation, you will find that the value of sinA is approximately 0.579.
Step 6: Since 90º≤A≤180º, we need to find the value of A using the inverse sine function, which will give us the angle in radians.
A = sin^(-1)(0.579)
Using a calculator, you will find that A is approximately 0.611 radians.
Therefore, the solution for 90º≤A≤180º is approximately 0.611 radians.
Now let's move on to solving the second question:
To solve 3tan^2A = 2tanA + 1 for 0º≤A≤90º, we can follow these steps:
Step 1: Rearrange the equation to get it in quadratic form.
3tan^2A - 2tanA - 1 = 0
Step 2: Factor the quadratic equation.
(3tanA - 1)(tanA + 1) = 0
Step 3: Set each factor equal to zero and solve for tanA.
3tanA - 1 = 0 --> tanA = 1/3
tanA + 1 = 0 --> tanA = -1
Step 4: Since we are looking for the values of A in the range 0º≤A≤90º, we can discard the negative tangent value (-1) because it is not within the specified range.
Step 5: Calculate the inverse tangent (arctan) of 1/3 to find the value of A.
A = arctan(1/3)
Using a calculator, you will find that A is approximately 18.4º.
Therefore, the solution for 0º≤A≤90º is approximately 18.4 degrees.