If tan(x)= -(3/4) and x is obtuse, evaluate sin(2x).
How do I get started on this question?
This angle x is obtuse, between 90 and 180 degrees. Therefore 2x is between 180 and 360.
tan x = -3/4
height = 3
base = 4
hypotenuse = 5 (3,4, 5 right triangle)
so
sin x = 3/5
cos x = -4/5
and we know sin 2x = 2 sin x cos x
so
sin 2x = 2(3/5)(-4/5)
sin 2x = -24/25
Angles with negative values of the tangent are in the second or fourth quadrant. If it is obtuse (between 90 and 180 degrees), it is in the second quadrant. The supplement of the angle is sin^-1 (3/5)= 36.87 degrees. (Think of a 3,4,5 right reference triangle). The angle x is therefore 143.13 degrees, and its sine is 3/5. Its cosine is -4/5. Use sin 2x = 2 sin x cos x to get the sine of twice the angle.
Sin (2x) = 2(3/5)(-4/5) = -24/25 = -0.96. It is in the fourth quadrant.
Well, first things first, we need to find the value of x in order to evaluate sin(2x). Since we know tan(x) = -(3/4), we can use the inverse tangent (or arctan) function to find x. But be careful, since x is obtuse, we will need to find the reference angle and then adjust it accordingly. Math can be quite the contortionist sometimes!
To start solving this question, you need to determine the reference angle (between 0 and 90 degrees) associated with the given value of tangent. The formula for tangent is tan(x) = opposite/adjacent.
Given that tan(x) = -(3/4), we can assign the opposite side as -3 and the adjacent side as 4. Since the value is negative, we can place the point in the third quadrant, where both x and y values are negative.
From this, we can calculate the hypothenuse using the Pythagorean theorem:
h = √(opposite² + adjacent²)
h = √((-3)² + 4²)
h = √(9 + 16)
h = √25
h = 5
Now that we have the lengths of all sides, we can determine the sine of the angle.
sin(x) = opposite/hypothenuse
sin(x) = -3/5
Next, we use the double angle identity for sine, which states that sin(2x) = 2*sin(x)*cos(x).
Since x is obtuse and located in the third quadrant, both sine and cosine are negative. Therefore, we can write the equation as:
sin(2x) = 2*(-3/5)*(-4/5)
Now we can simplify:
sin(2x) = 6/5 * 4/5
sin(2x) = 24/25
Thus, sin(2x) is equivalent to 24/25.
To solve this question, we need to find the value of sin(2x) given that tan(x) = -(3/4) and x is obtuse.
To get started, we can use the relationship between tangent and sine by using the identity:
tan(x) = sin(x)/cos(x)
Since we are given tan(x) = -(3/4), we can rewrite this as:
-(3/4) = sin(x)/cos(x)
Now, let's find the values of sin(x) and cos(x) using the given information. Since tan(x) is negative and x is obtuse, we know that the sine of x is negative, and the cosine of x is positive.
To find sin(x), we need to find the ratio of the opposite side to the hypotenuse in a right triangle. Let's create a right triangle where the opposite side is -3 and the adjacent side is 4. Since the sine is negative, we can place the triangle in the fourth quadrant.
4
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Therefore, sin(x) = -3/5.
To find cos(x), we can use Pythagorean theorem to find the length of the hypotenuse of the right triangle:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = (-3)^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25
hypotenuse = 5
Since cos(x) is positive and in the fourth quadrant, cos(x) = 4/5.
Now, let's evaluate sin(2x) using the double-angle identity for sine:
sin(2x) = 2*sin(x)*cos(x)
Plugging in the values we found earlier:
sin(2x) = 2*(-3/5)*(4/5)
Simplifying further:
sin(2x) = -24/25
Therefore, sin(2x) = -24/25.