cotx = -2 and 0 less than or equal to x less than or equal to pi
Find sin2x and cos2x
How to get the answers of sin2x = -4/5, cos2x = 3/5?
If cot x = -2 and x is in the first or second quadrant, it must be in the second quadrant, with a horizontal (x') coordinate -2 times the vertical (y') coordinate. The cosine of x is 2/sqrt 5 and the sine is -1/sqrt 5. You get that from the Pythagorean theorem and the fact that you are in the second quadrant.
Sin 2x = 2 sin x cos x = 2*(-2)/5 = -4/5
Cos 2x = cos^2 x - sin^2 x = 4/5 - 1/5 = 3/5
To find the values of sin(2x) and cos(2x), we need to use the given information that cot(x) = -2 and the range of x is from 0 to π.
First, let's recall the trigonometric identities:
1. cot(x) = cos(x)/sin(x)
2. sin^2(x) + cos^2(x) = 1
3. cos(2x) = cos^2(x) - sin^2(x)
4. sin(2x) = 2sin(x)cos(x)
Now, let's go step by step to find the values of sin(2x) and cos(2x):
Step 1: Find the values of sin(x) and cos(x).
Given that cot(x) = -2, we can use the identity cot(x) = cos(x)/sin(x) and substitute the value of cot(x) to get:
-2 = cos(x)/sin(x)
To solve for cos(x), we can multiply both sides of the equation by sin(x):
-2sin(x) = cos(x)
Now, we have an expression for cos(x) in terms of sin(x).
Step 2: Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1.
We can substitute the expression for cos(x) from step 1 into the Pythagorean identity:
sin^2(x) + (-2sin(x))^2 = 1
Expanding the equation:
sin^2(x) + 4sin^2(x) = 1
Combining like terms:
5sin^2(x) = 1
Divide both sides by 5:
sin^2(x) = 1/5
Take the square root of both sides to find sin(x):
sin(x) = ±√(1/5)
Since x is in the range of 0 ≤ x ≤ π, we take the positive square root:
sin(x) = √(1/5) = 1/√5 = 1/√5 * √5/√5 = √5/5
Step 3: Find cos(x) using the expression from step 1:
cos(x) = -2sin(x) = -2 * (√5/5) = -2√5/5
Step 4: Find cos(2x) using the identity cos(2x) = cos^2(x) - sin^2(x):
cos(2x) = cos^2(x) - sin^2(x)
Substituting the values found in step 2:
cos(2x) = (-2√5/5)^2 - (√5/5)^2
Simplifying:
cos(2x) = 4/5 - 1/5
cos(2x) = 3/5
Step 5: Find sin(2x) using the identity sin(2x) = 2sin(x)cos(x):
sin(2x) = 2sin(x)cos(x)
Substituting the values found in step 2:
sin(2x) = 2 * (√5/5) * (-2√5/5)
Simplifying:
sin(2x) = -4/5
Therefore, the values of sin(2x) and cos(2x) when cot(x) = -2 and 0 ≤ x ≤ π are sin(2x) = -4/5 and cos(2x) = 3/5.