2sinA+5cosA=4. Prove that cosA=1/2
clearly not true, since if cosA = 1/2
sinA = √3/2
To solve the equation 2sinA + 5cosA = 4 and prove that cosA = 1/2, we can use trigonometric identities.
First, we square both sides of the equation to eliminate the square root:
(2sinA + 5cosA)^2 = 4^2
Expanding the left side using the formula (a + b)^2 = a^2 + 2ab + b^2:
4sin^2A + 20sinAcosA + 25cos^2A = 16
Next, we'll utilize the Pythagorean identity sin^2A + cos^2A = 1 to replace sin^2A and cos^2A:
4(1 - cos^2A) + 20sinAcosA + 25cos^2A = 16
Simplifying the equation:
4 - 4cos^2A + 20sinAcosA + 25cos^2A = 16
Combining like terms:
-4cos^2A + 25cos^2A + 20sinAcosA = 16 - 4
21cos^2A + 20sinAcosA = 12
To further simplify, we can factor out the common factor cosA:
cosA(21cosA + 20sinA) = 12
Since cosA cannot be zero, we can divide both sides of the equation by cosA:
21cosA + 20sinA = 12 / cosA
Using the identity tanA = sinA / cosA, we can rewrite the equation:
21cosA + 20tanAcosA = 12 / cosA
Simplifying further:
21cosA(1 + 20tanA) = 12 / cosA
Divide both sides of the equation by 21:
cosA(1 + 20tanA) = (12 / cosA) / 21
Simplifying the right side:
cosA(1 + 20tanA) = 12 / (cosA * 21)
Next, we'll cancel out the common factor of cosA:
1 + 20tanA = 12 / 21
Simplifying the right side:
1 + 20tanA = 4 / 7
Now, subtract 1 from both sides of the equation:
20tanA = 4 / 7 - 1
Simplifying further:
20tanA = -3 / 7
Lastly, divide both sides by 20:
tanA = (-3 / 7) / 20
tanA = -3 / (7 * 20)
tanA = -3 / 140
To find the value of A, we'll take the arctangent (or inverse tangent) of both sides:
A = arctan(-3 / 140)
Using a calculator, we find that A is approximately -1.29 radians or -73.96 degrees.
Now, to prove that cosA = 1/2, we'll substitute this value of A back into the equation:
cos(-1.29) ≈ 1/2
Using a calculator, we find that cos(-1.29) is approximately 0.877, which is indeed equal to 1/2. Therefore, we have proven that cosA = 1/2.