Which of the following counterexamples proves that sinxtanx=cosx is not a trigonometric identity? Select all that apply.
-2π
-3π
-3π/4
-π/4
method1:
simply substitute each value into the equation to each if
it satisfies.
If the given value satisfies the equation, it clearly cannot be
used as a counter-example.
If the given value does not satisfy the equation ........
method 2: actually solve the given equation to see which
values are solutions.
sinxtanx = cosx
sinx(sinx/cos) = cosx
sin^2 x = cos^2 x
cos^2 x - sin^2 x = 0
cos(2x) = 0
2x = ± π/2 , ± because of the symmetry of the cosine curve in the y-axis
x = ± π/4
since cos 2x has a period of π
+π/4 - π = -3π/4 is also solution
Clearly since -π/4 and -3π/4 are in your list and they satisfy
the equation they CANNOT be used as counterexamples.
To check if sin(x)tan(x) = cos(x) is a trigonometric identity, we can substitute different values of x and see if the equation holds true.
Let's substitute each of the given values and check the equation:
1. For x = -2π:
sin(-2π)tan(-2π) = cos(-2π)
0 * 0 = 1 (which is not true)
So, -2π is a counterexample.
2. For x = -3π:
sin(-3π)tan(-3π) = cos(-3π)
0 * 0 = 1 (which is not true)
So, -3π is a counterexample.
3. For x = -3π/4:
sin(-3π/4)tan(-3π/4) = cos(-3π/4)
-√2 * -∞ = -√2 (which is not true)
So, -3π/4 is a counterexample.
4. For x = -π/4:
sin(-π/4)tan(-π/4) = cos(-π/4)
-√2 * -1 = √2 (which is true)
So, -π/4 is not a counterexample.
Therefore, the counterexamples that prove that sin(x)tan(x) = cos(x) is not a trigonometric identity are:
-2π, -3π, and -3π/4.
To determine whether sin(x)tan(x) = cos(x) is a trigonometric identity or not, we can substitute the given angles and check if the equation holds true.
Let's go through each option and substitute the values:
1. For -2π:
sin(-2π)tan(-2π) = cos(-2π)
sin(2π)tan(2π) = cos(2π)
Using the periodicity property of trigonometric functions, sin(2π) and cos(2π) both evaluate to 0, but tan(2π) is undefined. Therefore, this option does not hold true for the equation sin(x)tan(x) = cos(x), so -2π is a counterexample.
2. For -3π:
sin(-3π)tan(-3π) = cos(-3π)
sin(3π)tan(3π) = cos(3π)
Similarly, sin(3π) and cos(3π) both evaluate to 0, but tan(3π) is undefined. Hence, -3π is a counterexample.
3. For -3π/4:
sin(-3π/4)tan(-3π/4) = cos(-3π/4)
(-√2/2)(1) = (√2/2)
Upon simplifying, we see that the left and right sides of the equation are not equal. Hence, -3π/4 is also a counterexample.
4. For -π/4:
sin(-π/4)tan(-π/4) = cos(-π/4)
(-√2/2)(-1) = (√2/2)
After simplifying, we find that the left and right sides of the equation are indeed equal. Therefore, -π/4 is not a counterexample.
In summary, the counterexamples for the equation sin(x)tan(x) = cos(x) are -2π, -3π, and -3π/4.