SinA+ cosA=17/3 find sinA-cosA
sinA + cosA = 17/3
square both sides
sin^2 A + 2sinAcosA + cos^2 = 289/9
1 + 2sinAcosA = 289/9
2sinAcosA = 280/9
sinAcosA = 140/9
let x = sinA - cosA
x^2 = sin^2 A - 2sinAcosA + cos^2 A
= 1 - 2sinAcosA
= 1 - 280/9
= - 271/9
which is not possible, (x^2 cannot be negative)
Let's think about this.
y = sinA + cosA is a sinusoidal function with an amplitude of √(1^2 + 1^2) = √2
which is less than 17/3
sinA + cosA = 17/3 is not possible, I suspect a typo.
SinA+SinA=17/3
still not possible.
sinA cannot be greater than 1.
17/3 > 2, so sinA+sinA can never be 17/3, which is almost 6!
Better revisit the original problem.
To find the value of sinA - cosA, we need to use the given equation, sinA + cosA = 17/3, and manipulate it to obtain the desired expression.
Step 1: Square both sides of the equation sinA + cosA = 17/3.
(sinA + cosA)^2 = (17/3)^2
sin^2A + 2sinAcosA + cos^2A = 289/9
Step 2: Since sin^2A + cos^2A = 1 (a well-known trigonometric identity), we can substitute 1 in place of sin^2A + cos^2A in the equation obtained from Step 1.
1 + 2sinAcosA = 289/9
Step 3: Rearrange the equation to isolate sinAcosA.
2sinAcosA = 289/9 - 1
2sinAcosA = 289/9 - 9/9
2sinAcosA = 280/9
Step 4: Simplify the equation further by dividing both sides by 2.
sinAcosA = 280/9 / 2
sinAcosA = 140/9
Now, we have the value of sinAcosA. To find sinA - cosA, we'll need to make use of an additional identity:
sin(A - B) = sinAcosB - cosAsinB
In this case, we can write sinA - cosA as sin(A - A). Hence,
sin(A - A) = sinAcosA - cosAsinA
Since sin(A - A) is equivalent to sin0, which is 0, we substitute 0 for sin(A - A) and sinAcosA for sinAcosA in the equation:
0 = sinAcosA - cosAsinA
Rearranging this equation, we get:
- sinA = - cosAsinA
Dividing both sides by -1, we obtain:
sinA = cosAsinA
Now we can substitute sinA = cosAsinA in the equation sinA + cosA = 17/3:
cosAsinA + cosA = 17/3
Factoring out cosA, we have:
cosA(sinA + 1) = 17/3
Finally, we can find the value of sinA - cosA by substituting sinA = cosAsinA in the above equation:
cosA(cosAsinA + 1) = 17/3
cosA(cosA * cosA + 1) = 17/3
cosA(cos^2A + 1) = 17/3
Since we have only one equation with two variables (cosA and cos^2A), we can't directly solve for the value of sinA - cosA without any further information.