If sin 2 theta = (2/5) in Quadrant 2, find sin theta, cos theta, tan theta, and draw triangle theta.
To find sin(theta), cos(theta), and tan(theta) given that sin(2theta) = 2/5 in Quadrant 2, we can use the double-angle identity for sine:
sin(2theta) = 2sin(theta)cos(theta)
From this, we can deduce that:
2/5 = 2sin(theta)cos(theta)
Dividing both sides by 2, we get:
sin(theta)cos(theta) = 1/5
Now, let's use the Pythagorean identity, sin²(theta) + cos²(theta) = 1.
Since we know that sin(theta)cos(theta) = 1/5, we can derive the values of sin(theta) and cos(theta):
(sin(theta))^2 + (cos(theta))^2 = 1
(sin(theta))^2 + (1/5)^2 = 1 [Substituting cos(theta) = 1/5]
(sin(theta))^2 + 1/25 = 1
(sin(theta))^2 = 1 - 1/25
(sin(theta))^2 = 24/25
Taking the square root of both sides, we find:
sin(theta) = ± √(24/25)
sin(theta) = ± (2√6)/5
Since we are in Quadrant 2, where sin(theta) is positive, we have:
sin(theta) = (2√6)/5
To find cos(theta), we can use the Pythagorean identity:
(sin(theta))^2 + (cos(theta))^2 = 1
[(2√6)/5]^2 + (cos(theta))^2 = 1
4/25 + (cos(theta))^2 = 1
(cos(theta))^2 = 1 - 4/25
(cos(theta))^2 = 21/25
Taking the square root of both sides, we find:
cos(theta) = ± √(21)/5
Since we are in Quadrant 2, where cos(theta) is negative, we have:
cos(theta) = -√(21)/5
Now, we can find tan(theta) by dividing sin(theta) by cos(theta):
tan(theta) = sin(theta)/cos(theta)
tan(theta) = [(2√6)/5] / [-√(21)/5]
tan(theta) = -2√6 / √21
To draw the triangle theta, we can use the values we found for sin(theta) and cos(theta) to construct a right triangle. Based on the values we have:
sin(theta) = (2√6)/5
cos(theta) = -√21/5
We can assign a hypotenuse length of 5, as it is a common factor. Then, we can find the remaining side lengths using the Pythagorean theorem:
Let the adjacent side be x, then:
(sin(theta))^2 + (cos(theta))^2 = 1
[(2√6)/5]^2 + (-√21/5)^2 = 1
(4/25)(6) + (21/25) = 1
24/25 + 21/25 = 1
45/25 = 1
This simplifies to:
24 + 21 = 25
Therefore, we have:
(sin(theta))^2 = 24/25
(cos(theta))^2 = 21/25
Taking the square root of both sides, we get:
sin(theta) = 2√6/5
cos(theta) = -√21/5
Now, we can draw the triangle theta. Label the hypotenuse as 5, the side opposite theta as 2√6, and the adjacent side as -√21:
|
| 2√6
|
|_____
5
To find the values of sin theta, cos theta, and tan theta, we can use the given information that sin 2 theta = (2/5) in Quadrant 2.
First, let's recall the double angle formula for sine:
sin 2 theta = 2 * sin theta * cos theta
Using this formula, we can rewrite the given equation as:
2 * sin theta * cos theta = (2/5)
Now, let's solve for sin theta and cos theta separately:
sin theta * cos theta = (2/5) / 2
sin theta * cos theta = 1/5
Next, we use the Pythagorean identity to find the value of sin theta:
sin^2 theta + cos^2 theta = 1
Since sin theta * cos theta = 1/5, we can square both sides of this equation:
(sin theta * cos theta)^2 = (1/5)^2
(sin theta)^2 * (cos theta)^2 = 1/25
Now, let's rearrange the equation:
(cos theta)^2 = 25 / (sin theta)^2
cos^2 theta = 25 / (1 - cos^2 theta)
Substituting (1 - cos^2 theta) = sin^2 theta from the Pythagorean identity:
cos^2 theta = 25 / (1 - (25 / (sin theta)^2))
cos^2 theta = 25 sin^2 theta / (sin^2 theta - 25)
Finally, let's solve for cos theta:
cos theta = ± sqrt(25 sin^2 theta / (sin^2 theta - 25))
Notice that the values of sin theta and cos theta cannot be determined uniquely from the given information, as we have both positive and negative solutions. To proceed in finding the exact values, we need additional information.
As for the value of tan theta, we can use the formula:
tan theta = sin theta / cos theta
Substituting the values of sin theta and cos theta derived earlier, we get:
tan theta = (sin theta) / (± sqrt(25 sin^2 theta / (sin^2 theta - 25)))
Again, the exact value of tan theta cannot be determined uniquely without additional information.
To draw the triangle theta, we can make use of the information given in Quadrant 2. In this quadrant, the values of sine and cosine are positive, while the values of tangent are negative.
Start by drawing a coordinate system with the y-axis representing sine and the x-axis representing cosine. Since sin theta = 2/5, locate the point (0, 2/5) on the y-axis. To determine the cosine, you can use the Pythagorean identity to calculate cos theta.
Once you have the coordinates of the point (cos theta, sin theta), connect this point to the origin (0, 0) and the x-axis to form the triangle theta.