Question

Find the exact value of cosine left parenthesis alpha minus beta right parenthesis​, given that sine alpha equals startfraction 35 over 37 endfraction and cosine beta equals five thirteenths ​, with alpha in quadrant ii and beta in quadrant iv.

Answer 1

The exact value of cos(α-β) is found using trigonometric identities. Calculating cos(α) and sin(β), the result is -480/481, showcasing precise calculations in trigonometry.

To find the exact value of cos(α-β), we first need to calculate cos(α) and sin(β) using the given information.

1. Calculate cos(α):

- cos(α) = -√(1 - sin²(α))

- cos(α) = -√(1 - (35/37)²)

- cos(α) = -√(1 - 1225/1369)

- cos(α) = -√(144/1369)

- cos(α) = -12/37

2. Calculate sin(β):

- sin(β) = -√(1 - cos²(β))

- sin(β) = -√(1 - (5/13)²)

- sin(β) = -√(1 - 25/169)

- sin(β) = -√(144/169)

- sin(β) = -12/13

Now that we have cos(α) and sin(β), we can use the angle difference identity:

cos(α-β) = cos(α)cos(β) + sin(α)sin(β)

Substituting the values:

cos(α-β) = (-12/37)(5/13) + (35/37)(-12/13)

cos(α-β) = -60/481 - 420/481

cos(α-β) = -480/481

Therefore, the exact value of cos(α-β) is -480/481.

Answer 2

-480/481

Explanation:

You want the exact value of cos(α-β), given that sin(α) = 35/37 and cos(β) = 5/13, with α in the 2nd quadrant and β in the 4th quadrant.

Trig identities

The relevant trig identities are ...

cos(α-β) = cos(α)cos(β) +sin(α)sin(β)

cos(α) = -√(1 -sin²(α))

sin(β) = -√(1 -cos²(β))

Using these last two identities, we can find the values necessary to fill in the first identity.

cos(α) = -√(1 -(35/37)²) = -√(37² -35²)/37 = -√144/37 = -12/37

sin(β) = -√(1 -(5/13)²) = -√(13² -5²)/13 = -√144/13 = -12/13

Now, the cosine of the angle difference is ...

cos(α-β) = (-12/37)(5/13) + (35/37)(-12/13) = (-12)(5+35)/(37·13)

cos(α-β) = -480/481

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