if 5cos A plus 3=0 and 180<A<360..without using a calculator and with the aid of a diagram,determine the value of 1.1 sinA plus cosA 1.2 tanA *sinA
5cosA+3 = 0 means that
cosA = -3/5
So, A is in QIII
sinA = y/r = -4/5
sinA+cosA = -4/5 + -3/5 = -7/5
tanA*sinA = 4/3 * -4/5 = -16/15
To determine the value of 1.1 sinA + cosA and 1.2 tanA * sinA given the equation 5cos A + 3 = 0 and the range of A (180° < A < 360°), we can follow these steps:
Step 1: Solve the equation 5cos A + 3 = 0
Subtract 3 from both sides:
5cos A = -3
Divide both sides by 5:
cos A = -3/5
Step 2: Use the Pythagorean identity to find sin A:
sin^2 A + cos^2 A = 1
sin^2 A = 1 - cos^2 A
sin^2 A = 1 - (-3/5)^2
sin^2 A = 1 - 9/25
sin^2 A = 16/25
sin A = ±√(16/25)
sin A = ±(4/5)
Since A lies in the range 180° < A < 360°, sin A is positive:
sin A = 4/5
Step 3: Calculate 1.1 sin A + cos A:
1.1 sin A + cos A = 1.1(4/5) + (-3/5)
1.1 sin A + cos A = 4.4/5 - 3/5
1.1 sin A + cos A = 1.4/5
1.1 sin A + cos A = 0.28
Step 4: Calculate 1.2 tan A * sin A:
tan A = sin A / cos A
tan A = (4/5) / (-3/5)
tan A = -4/3
1.2 tan A * sin A = 1.2(-4/3)(4/5)
1.2 tan A * sin A = -1.28
Therefore, 1.1 sin A + cos A ≈ 0.28 and 1.2 tan A * sin A ≈ -1.28.
To find the values of 1.1sin(A) + cos(A) and 1.2tan(A) * sin(A) using the given equation 5cos(A) + 3 = 0 and a diagram, you can follow these steps:
Step 1: Solve the equation 5cos(A) + 3 = 0 to find the value of cos(A).
Subtracting 3 from both sides:
5cos(A) = -3
Dividing both sides by 5:
cos(A) = -3/5
Step 2: Use the Pythagorean identity to find sin(A).
For any angle A, the Pythagorean identity states that sin^2(A) + cos^2(A) = 1.
Substituting the value of cos(A) found in step 1:
sin^2(A) + (-3/5)^2 = 1
sin^2(A) + 9/25 = 1
sin^2(A) = 1 - 9/25
sin^2(A) = 25/25 - 9/25
sin^2(A) = 16/25
Taking the square root of both sides:
sin(A) = ±√(16/25)
sin(A) = ±(√16)/(√25)
sin(A) = ±4/5
Since 180° < A < 360°, we are only interested in the values of sin(A) in the second quadrant (sin(A) > 0) because cosine is negative in the second quadrant.
Therefore, sin(A) = 4/5.
Step 3: Calculate 1.1sin(A) + cos(A):
Substituting the values of sin(A) and cos(A) into the expression:
1.1sin(A) + cos(A) = 1.1 * (4/5) + (-3/5)
1.1sin(A) + cos(A) = 4.4/5 - 3/5
1.1sin(A) + cos(A) = (4.4 - 3)/5
1.1sin(A) + cos(A) = 1.4/5
1.1sin(A) + cos(A) = 14/50
1.1sin(A) + cos(A) = 7/25
Therefore, the value of 1.1sin(A) + cos(A) is 7/25.
Step 4: Calculate 1.2tan(A) * sin(A):
Using the trigonometric identity tan(A) = sin(A)/cos(A):
1.2tan(A) * sin(A) = 1.2 * (sin(A)/cos(A)) * sin(A)
1.2tan(A) * sin(A) = 1.2 * (4/5) * (4/5)
1.2tan(A) * sin(A) = 1.2 * 16/25
1.2tan(A) * sin(A) = 19.2/25
Therefore, the value of 1.2tan(A) * sin(A) is 19.2/25.
By following these steps, you can determine the values of 1.1sin(A) + cos(A) and 1.2tan(A) * sin(A) without using a calculator and with the aid of a diagram.