Question Find all real numbers $x$ that satisfy the equation $\cos x = 15$ .

Studdy Solution

STEP 1

Assumptions
1. We are looking for real number solutions to the equation.
2. The variable $x$ represents a real number.
3. We will use the properties of the cosine function and the basic principles of algebra to solve the equation.

STEP 2

First, we need to isolate the cosine term on one side of the equation. We can do this by adding 8 to both sides of the equation.
$\cos x - 8 + 8 = -7 + 8$

STEP 3

Now, simplify the equation by combining like terms.
$\cos x = 1$

STEP 4

We need to find all real numbers $x$ such that the cosine of $x$ equals 1.

STEP 5

Recall that the cosine function has a maximum value of 1, and this occurs at integer multiples of $2\pi$ radians, where the cosine function equals 1.

STEP 6

Therefore, the general solution for $x$ where $\cos x = 1$ is given by:
$x = 2k\pi$
where $k$ is any integer.

STEP 7

We can express the solution set for $x$ as:
$x \in \{2k\pi \mid k \in \mathbb{Z}\}$
This represents all real numbers that satisfy the given equation.

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Question Find all real numbers $x$ that satisfy the equation $\cos x = 15$ .

Studdy Solution

STEP 1

Assumptions
1. We are looking for real number solutions to the equation.
2. The variable $x$ represents a real number.
3. We will use the properties of the cosine function and the basic principles of algebra to solve the equation.

STEP 2

First, we need to isolate the cosine term on one side of the equation. We can do this by adding 8 to both sides of the equation.
$\cos x - 8 + 8 = -7 + 8$

STEP 3

Now, simplify the equation by combining like terms.
$\cos x = 1$

STEP 4

We need to find all real numbers $x$ such that the cosine of $x$ equals 1.

STEP 5

Recall that the cosine function has a maximum value of 1, and this occurs at integer multiples of $2\pi$ radians, where the cosine function equals 1.

STEP 6

Therefore, the general solution for $x$ where $\cos x = 1$ is given by:
$x = 2k\pi$
where $k$ is any integer.

STEP 7

We can express the solution set for $x$ as:
$x \in \{2k\pi \mid k \in \mathbb{Z}\}$
This represents all real numbers that satisfy the given equation.

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Question Find all real numbers xxx that satisfy the equation cos⁡x=15\cos x = 15cosx=15.

Studdy Solution

STEP 1

Assumptions1. We are looking for real number solutions to the equation.2. The variable xxx represents a real number.3. We will use the properties of the cosine function and the basic principles of algebra to solve the equation.

STEP 2

First, we need to isolate the cosine term on one side of the equation. We can do this by adding 8 to both sides of the equation.cos⁡x−8+8=−7+8\cos x - 8 + 8 = -7 + 8cosx−8+8=−7+8

STEP 3

Now, simplify the equation by combining like terms.cos⁡x=1\cos x = 1cosx=1

STEP 4

We need to find all real numbers xxx such that the cosine of xxx equals 1.

STEP 5

Recall that the cosine function has a maximum value of 1, and this occurs at integer multiples of 2π2\pi2π radians, where the cosine function equals 1.

STEP 6

Therefore, the general solution for xxx where cos⁡x=1\cos x = 1cosx=1 is given by:x=2kπx = 2k\pix=2kπwhere kkk is any integer.

STEP 7

We can express the solution set for xxx as:x∈{2kπ∣k∈Z}x \in \{2k\pi \mid k \in \mathbb{Z}\}x∈{2kπ∣k∈Z}This represents all real numbers that satisfy the given equation.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Find all real numbers $x$ that satisfy the equation $\cos x = 15$ .

Assumptions
1. We are looking for real number solutions to the equation.
2. The variable $x$ represents a real number.
3. We will use the properties of the cosine function and the basic principles of algebra to solve the equation.

First, we need to isolate the cosine term on one side of the equation. We can do this by adding 8 to both sides of the equation.
$\cos x - 8 + 8 = -7 + 8$

Now, simplify the equation by combining like terms.
$\cos x = 1$

We need to find all real numbers $x$ such that the cosine of $x$ equals 1.

Recall that the cosine function has a maximum value of 1, and this occurs at integer multiples of $2\pi$ radians, where the cosine function equals 1.

Therefore, the general solution for $x$ where $\cos x = 1$ is given by:
$x = 2k\pi$
where $k$ is any integer.

We can express the solution set for $x$ as:
$x \in \{2k\pi \mid k \in \mathbb{Z}\}$
This represents all real numbers that satisfy the given equation.