Solved on Dec 06, 2023

Use sum/diff identities to find the value of $\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}$ . If undefined, write DNE.

STEP 1

Assumptions
1. We are working with angles in degrees.
2. We will use the sum and difference identities for sine and cosine.
3. The sum identity for sine is $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ .
4. The difference identity for cosine is $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ .
5. We are looking for the exact value of the expression.

STEP 2

Recognize that the given expression resembles the sum identity for sine.
$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

STEP 3

Identify the angles $\alpha$ and $\beta$ that will allow us to use the sum identity to simplify the expression.
Let $\alpha = 125^{\circ}$ and $\beta = 55^{\circ}$ .

STEP 4

Apply the sum identity for sine to the expression using the identified angles.
$\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ} = \sin(125^{\circ} + 55^{\circ})$

STEP 5

Calculate the sum of the angles.
$125^{\circ} + 55^{\circ} = 180^{\circ}$

STEP 6

Substitute the sum of the angles into the sine function.
$\sin(125^{\circ} + 55^{\circ}) = \sin 180^{\circ}$

STEP 7

Evaluate the sine of $180^{\circ}$ .
$\sin 180^{\circ} = 0$

STEP 8

Conclude that the exact value of the original expression is 0.
The exact value of the expression $\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}$ is 0.

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Use sum/diff identities to find the value of $\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}$ . If undefined, write DNE.

STEP 1

STEP 2

Recognize that the given expression resembles the sum identity for sine.
$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

STEP 3

Identify the angles $\alpha$ and $\beta$ that will allow us to use the sum identity to simplify the expression.
Let $\alpha = 125^{\circ}$ and $\beta = 55^{\circ}$ .

STEP 4

Apply the sum identity for sine to the expression using the identified angles.
$\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ} = \sin(125^{\circ} + 55^{\circ})$

STEP 5

Calculate the sum of the angles.
$125^{\circ} + 55^{\circ} = 180^{\circ}$

STEP 6

Substitute the sum of the angles into the sine function.
$\sin(125^{\circ} + 55^{\circ}) = \sin 180^{\circ}$

STEP 7

Evaluate the sine of $180^{\circ}$ .
$\sin 180^{\circ} = 0$

STEP 8

Conclude that the exact value of the original expression is 0.
The exact value of the expression $\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}$ is 0.

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Use sum/diff identities to find the value of sin⁡125∘cos⁡55∘+cos⁡125∘sin⁡55∘\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}sin125∘cos55∘+cos125∘sin55∘. If undefined, write DNE.

STEP 1

STEP 2

Recognize that the given expression resembles the sum identity for sine.sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \betasin(α+β)=sinαcosβ+cosαsinβ

STEP 3

Identify the angles α\alphaα and β\betaβ that will allow us to use the sum identity to simplify the expression.Let α=125∘\alpha = 125^{\circ}α=125∘ and β=55∘\beta = 55^{\circ}β=55∘.

STEP 4

Apply the sum identity for sine to the expression using the identified angles.sin⁡125∘cos⁡55∘+cos⁡125∘sin⁡55∘=sin⁡(125∘+55∘)\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ} = \sin(125^{\circ} + 55^{\circ})sin125∘cos55∘+cos125∘sin55∘=sin(125∘+55∘)

STEP 5

Calculate the sum of the angles.125∘+55∘=180∘125^{\circ} + 55^{\circ} = 180^{\circ}125∘+55∘=180∘

STEP 6

Substitute the sum of the angles into the sine function.sin⁡(125∘+55∘)=sin⁡180∘\sin(125^{\circ} + 55^{\circ}) = \sin 180^{\circ}sin(125∘+55∘)=sin180∘

STEP 7

Evaluate the sine of 180∘180^{\circ}180∘.sin⁡180∘=0\sin 180^{\circ} = 0sin180∘=0

STEP 8

Conclude that the exact value of the original expression is 0.The exact value of the expression sin⁡125∘cos⁡55∘+cos⁡125∘sin⁡55∘\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}sin125∘cos55∘+cos125∘sin55∘ is 0.

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Use sum/diff identities to find the value of $\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}$ . If undefined, write DNE.

Recognize that the given expression resembles the sum identity for sine.
$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

Identify the angles $\alpha$ and $\beta$ that will allow us to use the sum identity to simplify the expression.
Let $\alpha = 125^{\circ}$ and $\beta = 55^{\circ}$ .

Apply the sum identity for sine to the expression using the identified angles.
$\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ} = \sin(125^{\circ} + 55^{\circ})$

Calculate the sum of the angles.
$125^{\circ} + 55^{\circ} = 180^{\circ}$

Substitute the sum of the angles into the sine function.
$\sin(125^{\circ} + 55^{\circ}) = \sin 180^{\circ}$

Evaluate the sine of $180^{\circ}$ .
$\sin 180^{\circ} = 0$

Conclude that the exact value of the original expression is 0.
The exact value of the expression $\sin 125^{\circ} \cos 55^{\circ} + \cos 125^{\circ} \sin 55^{\circ}$ is 0.