Question

Find the exact value of the expressions cos(α+β),sin(α+β) and tan(α+β) under the following conditions sin(α)= 24/25,α lies in quadrant I, and sin(β)= 12/13,β lies in quadrant II

a. cos(α+β)=
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Answer 1

The exact value of
`cos(\alpha + \beta)` under the given conditions is -323/325.

To find the exact value of cos(α+β), we need to use the trigonometric identity and the given information.

Given sin(α) = 24/25 and α lies in quadrant I, we can use the Pythagorean identity to find
`cos(\alpha)`:

`cos^2(\alpha ) + sin^2(\alpha ) = 1`

`cos^2(\alpha) + (24/25)^2 = 1`

`cos^2(\alpha) + 576/625 = 1`

`cos^2(\alpha) = 1 - 576/625`

`cos^2(\alpha ) = 49/625`

Since alpha lies in quadrant I and
`cos(\alpha )` is positive, we take the positive square root:

`cos(\alpha ) = √((49/625))`

`cos(\alpha ) = 7/25`

Similarly, given
`sin(\beta ) = 12/13` and
`\beta` lies in quadrant II, we can find
`cos(\beta)`:

`cos^2(\beta) + sin^2(\beta) = 1`

`cos^2(\beta) + (12/13)^2 = 1`

`cos^2(\beta) + 144/169 = 1`

`cos^2(\beta) = 1 - 144/169`

`cos^2(\beta) = 25/169`

Since beta lies in quadrant II and
`cos(\beta)` is negative, we take the negative square root:

`cos(\beta) = -√(25/169)`

`cos(\beta) = -5/13`

Now, we can use the angle addition formula to find the value of
`cos(\alpha + \beta)`:

`cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)`

`cos(\alpha + \beta) = (7/25)(-5/13) - (24/25)(12/13)`

`cos(\alpha + \beta) = -35/325 - 288/325`

`cos(\alpha + \beta) = -323/325`

In conclusion,
`Cos(\alpha + \beta) = -323/325`.

Answer 2

The exact value of the expressions are cos(α + β) = -323/325, sin(α + β) = -36/325 and tan(α + β) = 36/323

How to determine the exact value of the expressions

From the question, we have the following parameters that can be used in our computation:

• sin(α)= 24/25, where α lies in quadrant I
• sin(β)= 12/13, where β lies in quadrant II

Using the following equation

sin²(x) + cos²(x) = 1

We have

cos²(x) = 1 - sin²(x)

cos²(α) = 1 - sin²(α)

This gives

cos²(α) = 1 - (24/25)²

cos²(α) = 49/625

α lies in quadrant I

So:

cos(α) = 7/25

Next, we have

cos²(b) = 1 - sin²(b)

This gives

cos²(β) = 1 - (12/13)²

cos²(β) = 25/169

b lies in quadrant II

So:

cos(β) = -5/13

Using the above as a guide, we have the following:

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

cos(α + β) = 7/25 * -5/13 - 24/25 * 12/13

cos(α + β) = -35/325 -288/325

cos(α + β) = -323/325

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

sin(α + β) = 24/25 * -5/13 + 7/25 * 12/13

sin(α + β) = -120/325 + 84/325

sin(α + β) = -36/325

tan(α + β) = sin(α + β)/cos(α + β)

tan(α + β) = (-36/325)/(-323/325)

Evaluate

tan(α + β) = 36/323

Hence, the exact value of the expressions are cos(α + β) = -323/325, sin(α + β) = -36/325 and tan(α + β) = 36/323