QuestionEvaluate $\cos \frac{\pi}{4}$ using special right triangles and simplify your answer.

Studdy Solution

STEP 1

Assumptions1. We are working in a unit circle, where the hypotenuse (radius) is1. . We are using the special right triangle45-45-90.
3. We are using the definition of cosine as the ratio of the adjacent side to the hypotenuse.

STEP 2

In a45-45-90 triangle, the angles are45 degrees,45 degrees, and90 degrees. This is equivalent to $\frac{\pi}{4}$ , $\frac{\pi}{4}$ , and $\frac{\pi}{2}$ in radians.

STEP 3

The sides of a45-45-90 triangle have a specific ratio. If the hypotenuse is1 (as in a unit circle), then the other two sides are of equal length, and their length is $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$ .

STEP 4

The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse.

STEP 5

In our case, the angle is $\frac{\pi}{4}$ (or45 degrees), and in a45-45-90 triangle, the lengths of the two sides (not the hypotenuse) are equal. Therefore, the cosine of this angle is the length of one side divided by the length of the hypotenuse.

STEP 6

Substitute the lengths of the side and the hypotenuse into the definition of cosine.
$\cos \frac{\pi}{4} = \frac{Side}{Hypotenuse} = \frac{\frac{\sqrt{2}}{2}}{1}$

STEP 7

implify the expression.
$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ So, $\cos \frac{\pi}{4}$ equals $\frac{\sqrt{2}}{2}$ .

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QuestionEvaluate $\cos \frac{\pi}{4}$ using special right triangles and simplify your answer.

Studdy Solution

STEP 1

Assumptions1. We are working in a unit circle, where the hypotenuse (radius) is1. . We are using the special right triangle45-45-90.
3. We are using the definition of cosine as the ratio of the adjacent side to the hypotenuse.

STEP 2

In a45-45-90 triangle, the angles are45 degrees,45 degrees, and90 degrees. This is equivalent to $\frac{\pi}{4}$ , $\frac{\pi}{4}$ , and $\frac{\pi}{2}$ in radians.

STEP 3

The sides of a45-45-90 triangle have a specific ratio. If the hypotenuse is1 (as in a unit circle), then the other two sides are of equal length, and their length is $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$ .

STEP 4

The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse.

STEP 5

In our case, the angle is $\frac{\pi}{4}$ (or45 degrees), and in a45-45-90 triangle, the lengths of the two sides (not the hypotenuse) are equal. Therefore, the cosine of this angle is the length of one side divided by the length of the hypotenuse.

STEP 6

Substitute the lengths of the side and the hypotenuse into the definition of cosine.
$\cos \frac{\pi}{4} = \frac{Side}{Hypotenuse} = \frac{\frac{\sqrt{2}}{2}}{1}$

STEP 7

implify the expression.
$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ So, $\cos \frac{\pi}{4}$ equals $\frac{\sqrt{2}}{2}$ .

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QuestionEvaluate cos⁡π4\cos \frac{\pi}{4}cos4π​ using special right triangles and simplify your answer.

Studdy Solution

STEP 1

Assumptions1. We are working in a unit circle, where the hypotenuse (radius) is1. . We are using the special right triangle45-45-90.3. We are using the definition of cosine as the ratio of the adjacent side to the hypotenuse.

STEP 2

In a45-45-90 triangle, the angles are45 degrees,45 degrees, and90 degrees. This is equivalent to π4\frac{\pi}{4}4π​, π4\frac{\pi}{4}4π​, and π2\frac{\pi}{2}2π​ in radians.

STEP 3

The sides of a45-45-90 triangle have a specific ratio. If the hypotenuse is1 (as in a unit circle), then the other two sides are of equal length, and their length is 12\frac{1}{\sqrt{2}}2​1​ or 22\frac{\sqrt{2}}{2}22​​.

STEP 4

The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse.

STEP 5

In our case, the angle is π4\frac{\pi}{4}4π​ (or45 degrees), and in a45-45-90 triangle, the lengths of the two sides (not the hypotenuse) are equal. Therefore, the cosine of this angle is the length of one side divided by the length of the hypotenuse.

STEP 6

Substitute the lengths of the side and the hypotenuse into the definition of cosine.cos⁡π4=SideHypotenuse=221\cos \frac{\pi}{4} = \frac{Side}{Hypotenuse} = \frac{\frac{\sqrt{2}}{2}}{1}cos4π​=HypotenuseSide​=122​​​

STEP 7

implify the expression.cos⁡π4=22\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}cos4π​=22​​So, cos⁡π4\cos \frac{\pi}{4}cos4π​ equals 22\frac{\sqrt{2}}{2}22​​.

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QuestionEvaluate $\cos \frac{\pi}{4}$ using special right triangles and simplify your answer.

Assumptions1. We are working in a unit circle, where the hypotenuse (radius) is1. . We are using the special right triangle45-45-90.
3. We are using the definition of cosine as the ratio of the adjacent side to the hypotenuse.

In a45-45-90 triangle, the angles are45 degrees,45 degrees, and90 degrees. This is equivalent to $\frac{\pi}{4}$ , $\frac{\pi}{4}$ , and $\frac{\pi}{2}$ in radians.

The sides of a45-45-90 triangle have a specific ratio. If the hypotenuse is1 (as in a unit circle), then the other two sides are of equal length, and their length is $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$ .

In our case, the angle is $\frac{\pi}{4}$ (or45 degrees), and in a45-45-90 triangle, the lengths of the two sides (not the hypotenuse) are equal. Therefore, the cosine of this angle is the length of one side divided by the length of the hypotenuse.

Substitute the lengths of the side and the hypotenuse into the definition of cosine.
$\cos \frac{\pi}{4} = \frac{Side}{Hypotenuse} = \frac{\frac{\sqrt{2}}{2}}{1}$

implify the expression.
$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ So, $\cos \frac{\pi}{4}$ equals $\frac{\sqrt{2}}{2}$ .