cos A = -4/5 for angle A in Quadrant II, find sin 2A
cosA = -4/5 = -0.80,
A = 143.1deg.
sin(2A) = 2*sinA*cosA,
sin2A = 2*sin(143.1)*cos(143.1 =
-0.9603.
Solution using Exact values.
cosA = -4/5 = x/r.
(-4)^2 + Y^2 = 5^2,
16 + Y^2 = 25,
Y^2 = 25 - 16 = 9,
Y = 3.
SinA = Y/r = 3/5.
sin2A = 2 * 3/5 * -4/5 = -24/25.
To find sin 2A, we need to first find sin A and cos 2A using the given information.
Given that cos A = -4/5, we can use the Pythagorean identity to find sin A:
sin^2 A + cos^2 A = 1
sin^2 A + (-4/5)^2 = 1
sin^2 A + 16/25 = 1
sin^2 A = 1 - 16/25
sin^2 A = 9/25
Taking the square root of both sides, we have:
sin A = ±√(9/25)
Since angle A is in Quadrant II, where sin A is positive, we have:
sin A = √(9/25)
Simplifying further:
sin A = 3/5
Now, we can find cos 2A using the double-angle identity:
cos 2A = cos^2 A - sin^2 A
Using the values we found earlier:
cos 2A = (-4/5)^2 - (3/5)^2
cos 2A = 16/25 - 9/25
cos 2A = 7/25
Finally, we can find sin 2A using another trigonometric identity:
sin 2A = 2 sin A cos A
Substituting the values we found earlier:
sin 2A = 2 (3/5) (7/25)
sin 2A = 6/5 * 7/25
sin 2A = 42/125
Therefore, sin 2A is equal to 42/125.
To find sin 2A, we need to know the value of cos 2A. We can use the double-angle formula for cosine to find cos 2A.
The double-angle formula for cosine is: cos 2A = cos^2(A) - sin^2(A)
Step 1: Find sin A
Since cos A = -4/5, we can use the Pythagorean identity to find sin A:
sin A = ± √(1 - cos^2 A)
= ± √(1 - (-4/5)^2)
= ± √(1 - 16/25)
= ± √(9/25)
= ± 3/5
Since angle A is in Quadrant II, sin A must be positive. Therefore, sin A = 3/5.
Step 2: Find cos^2 A
Using the given value of cos A = -4/5:
cos^2 A = (-4/5)^2
= 16/25
Step 3: Substitute the values into the double-angle formula for cosine:
cos 2A = cos^2(A) - sin^2(A)
= 16/25 - (3/5)^2
= 16/25 - 9/25
= 7/25
Step 4: Find sin 2A
Using the Pythagorean identity, sin^2 2A + cos^2 2A = 1:
(7/25)^2 + sin^2 2A = 1
sin^2 2A = 1 - (7/25)^2
= 1 - 49/625
= 576/625
Taking the square root of both sides to solve for sin 2A:
sin 2A = ± √(576/625)
= ± (24/25)
Since angle A is in Quadrant II, sin 2A must be positive. Therefore, sin 2A = 24/25.
So, sin 2A = 24/25.