Given that Ο < π₯ < 2Ο and π‘ππ(π₯) = 3/4, determine the exact value of πππ 2π₯. Show all work including a diagram, special triangles, CAST rules and related acute angles.
You know that tan ΞΈ = y/x, so x = 4 and y = 3
we also know that x^2 + y^2 = r^2
so r = 5
the tangent is positive in I and III, so
in I, tanx = 3/4, sinx = 3/5 and cosx = 4/5
cos 2x = cos^2 x - sin^2 x = 16/25 - 9/25 = 7/25
since we are squaring both sinx and cosx
the same would be true for quad III, so
cos 2x = 7/25
draw a triangle with legs 3 and 4, and you can see that since you are in QIII, that means that
sinx = -3/5 and cosx = -4/5
cos2x = 2cos^2x - 1 = 2 * 16/25 - 1 = 7/25
To determine the exact value of cos(2x), we can use the identity:
cos(2x) = cos^2(x) - sin^2(x)
First, we need to find the value of sin(x) using the given equation tan(x) = 3/4.
Using the definition of tan (tan(x) = sin(x) / cos(x)), we have:
3/4 = sin(x) / cos(x)
Cross-multiplying gives us:
4 * sin(x) = 3 * cos(x)
Dividing both sides by cos(x), we get:
sin(x) / cos(x) = 3/4
Using the Pythagorean identity (sin^2(x) + cos^2(x) = 1), we know that:
sin^2(x) = 1 - cos^2(x)
Substituting the value of sin(x) from the previous equation, we can solve for cos(x):
(3/4)^2 = 1 - cos^2(x)
9/16 = 1 - cos^2(x)
Rearranging the equation, we get:
cos^2(x) = 1 - 9/16
cos^2(x) = 16/16 - 9/16
cos^2(x) = 7/16
Taking the square root of both sides, we obtain:
cos(x) = Β±β(7/16)
Since Ο < x < 2Ο, x lies in the second quadrant where cos(x) is negative. Therefore, we take the negative root:
cos(x) = -β(7/16)
We can relate the given interval to the unit circle as follows:
- Ο < x < 2Ο means 180Β° < x < 360Β°
Using CAST (where C stands for cos, A for all trigonometric functions, S for sin, T for tan):
- In the second quadrant, sin and tan are positive, cos is negative.
- In the third and fourth quadrants, all trigonometric functions are negative.
We can draw a diagram of the unit circle in the second quadrant:
. (unit circle)
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In the second quadrant, the reference angle is Ξ± = 180Β° - x.
Using the special triangles (45-45-90 and 30-60-90 triangles), we can find the value of cos(x) and sin(x). For reference, the 45-45-90 triangle has sides in the ratio 1:1:β2, and the 30-60-90 triangle has sides in the ratio 1:β3:2.
Since sin(x) = 3/4, we can find cos(x):
sin(x) = 3/4
Using the Pythagorean identity (sin^2(x) + cos^2(x) = 1), we have:
(3/4)^2 + cos^2(x) = 1
9/16 + cos^2(x) = 1
cos^2(x) = 1 - 9/16
cos^2(x) = 16/16 - 9/16
cos^2(x) = 7/16
Taking the square root of both sides, we get:
cos(x) = Β±β(7/16)
Since x lies in the second quadrant, where cos(x) is negative:
cos(x) = -β(7/16)
Now, to find cos(2x), we can use the double angle identity:
cos(2x) = cos^2(x) - sin^2(x)
Substituting the values we found:
cos(2x) = (-β(7/16))^2 - (3/4)^2
cos(2x) = 7/16 - 9/16
cos(2x) = -2/16
Simplifying, we get:
cos(2x) = -1/8
Therefore, the exact value of cos(2x) is -1/8.
To determine the exact value of cos(2x), we need to use a combination of trigonometric identities, special triangles, and the CAST rule.
1. First, let's draw a diagram to represent the given information. We have a range of values for x, Ο < x < 2Ο, which falls in the second quadrant.
Ο βββββββββ€βββ x βββββ€ββββββ 2Ο
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ββββββββββββββ
2. Next, let's determine the related acute angles. In the second quadrant, we have the acute angle ΞΈ, formed between the positive x-axis and the terminal side of x (tan(x) = 3/4). To find ΞΈ, we can use the inverse tangent function (tan^(-1)).
ΞΈ = tan^(-1)(3/4) β 36.87Β°
3. Now, we need to determine the reference angle for x, which is the angle formed between the terminal side of x and the x-axis. Since the reference angle is always positive, we take the complement of ΞΈ (90Β° - ΞΈ).
Reference angle = 90Β° - 36.87Β° β 53.13Β°
4. Using the reference angle, we can use the special triangle (known as the 3-4-5 right triangle) to find the values of sin(ΞΈ) and cos(ΞΈ).
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90Β°
In this triangle, sin(ΞΈ) = 3/5 and cos(ΞΈ) = 4/5.
5. Now that we have the value of cos(ΞΈ), we can use the double-angle formula, which states that cos(2ΞΈ) = cos^2(ΞΈ) - sin^2(ΞΈ).
cos(2ΞΈ) = (cos(ΞΈ))^2 - (sin(ΞΈ))^2
= (4/5)^2 - (3/5)^2
= 16/25 - 9/25
= 7/25
Therefore, the exact value of cos(2x) is 7/25.