Given that π < 𝑥 < 2π and 𝑡𝑎𝑛(𝑥) = 3/4, determine the exact value of 𝑐𝑜𝑠2𝑥. Show all work including a diagram, special triangles, CAST rules and related acute angles.

Question

Given that π < 𝑥 < 2π and 𝑡𝑎𝑛(𝑥) = 3/4, determine the exact value of 𝑐𝑜𝑠2𝑥. Show all work including a diagram, special triangles, CAST rules and related acute angles.

Answer 1

You know that tan θ = 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

Answer 2

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

Answer 3

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) = ±√(7/16)

Since π < x < 2π, x lies in the second quadrant where cos(x) is negative. Therefore, we take the negative root:

cos(x) = -√(7/16)

We can relate the given interval to the unit circle as follows:
- π < x < 2π means 180° < x < 360°

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)
.
.
.___________.
| |
| |
| |
| |
| |
| |
|___________________|

In the second quadrant, the reference angle is α = 180° - 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:√2, and the 30-60-90 triangle has sides in the ratio 1:√3:2.

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) = ±√(7/16)

Since x lies in the second quadrant, where cos(x) is negative:

cos(x) = -√(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) = (-√(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.

Answer 4

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, π < x < 2π, which falls in the second quadrant.

π ────────┤─── x ────┤────── 2π
| |
└────────────┘

2. Next, let's determine the related acute angles. In the second quadrant, we have the acute angle θ, formed between the positive x-axis and the terminal side of x (tan(x) = 3/4). To find θ, we can use the inverse tangent function (tan^(-1)).

θ = tan^(-1)(3/4) ≈ 36.87°

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 θ (90° - θ).

Reference angle = 90° - 36.87° ≈ 53.13°

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(θ) and cos(θ).

|\
| \
4 | \ 3
| \
|____\
90°

In this triangle, sin(θ) = 3/5 and cos(θ) = 4/5.

5. Now that we have the value of cos(θ), we can use the double-angle formula, which states that cos(2θ) = cos^2(θ) - sin^2(θ).

cos(2θ) = (cos(θ))^2 - (sin(θ))^2
= (4/5)^2 - (3/5)^2
= 16/25 - 9/25
= 7/25

Therefore, the exact value of cos(2x) is 7/25.