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Math  /  Trigonometry

Question34sin⁡(x+5.793)=34(cos⁡□)34 \sin (x+5.793) = 34 \left( \cos \square \right)
Hello there! It seems we have a math problem involving the verification of a trigonometric identity using the sum formula for sine. However, it looks like there is some missing information. Could you provide the result from part (a) or any specific details you have for this problem? This will help me guide you through the solution effectively.
34 \sin(x + 0.493)

Studdy Solution

STEP 1

1. We are given the equation 34sin⁡(x+5.793)=34cos⁡(□) 34 \sin(x + 5.793) = 34 \cos(\square) .
2. We need to verify this equation using the sum formula for sine.
3. The problem suggests using trigonometric identities to simplify or verify the expression.

STEP 2

1. Simplify the given equation.
2. Apply the sum formula for sine.
3. Verify the trigonometric identity.

STEP 3

First, simplify the given equation by dividing both sides by 34:
34sin⁡(x+5.793)=34cos⁡(□) 34 \sin(x + 5.793) = 34 \cos(\square)
sin⁡(x+5.793)=cos⁡(□) \sin(x + 5.793) = \cos(\square)

STEP 4

Recall the identity for cosine in terms of sine:
cos⁡(θ)=sin⁡(π2−θ) \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)
We need to express cos⁡(□)\cos(\square) in terms of sine:
cos⁡(□)=sin⁡(π2−□) \cos(\square) = \sin\left(\frac{\pi}{2} - \square\right)

STEP 5

Set the expression for sine equal to the expression for cosine:
sin⁡(x+5.793)=sin⁡(π2−□) \sin(x + 5.793) = \sin\left(\frac{\pi}{2} - \square\right)

STEP 6

For the equality sin⁡(A)=sin⁡(B)\sin(A) = \sin(B) to hold, A A must be equal to B B or A=π−B A = \pi - B . Thus, we have:
x+5.793=π2−□ x + 5.793 = \frac{\pi}{2} - \square
or
x+5.793=π−(π2−□) x + 5.793 = \pi - \left(\frac{\pi}{2} - \square\right)

STEP 7

Solve for □\square in both scenarios:
1. x+5.793=π2−□ x + 5.793 = \frac{\pi}{2} - \square
□=π2−(x+5.793) \square = \frac{\pi}{2} - (x + 5.793)
2. x+5.793=π−(π2−□) x + 5.793 = \pi - \left(\frac{\pi}{2} - \square\right)
x+5.793=π2+□ x + 5.793 = \frac{\pi}{2} + \square
□=x+5.793−π2 \square = x + 5.793 - \frac{\pi}{2}
The expression □\square can be either π2−(x+5.793)\frac{\pi}{2} - (x + 5.793) or x+5.793−π2x + 5.793 - \frac{\pi}{2}.

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