Question

cos alpha equals 7 over 25 and sin beta equals 4 over 5. Both angles are in Quadrant I. Find cos left parenthesis alpha minus beta right parenthesis

Answer 1

`(117)/(125)`

Explanation:

We are given that:

`cos\alpha = (7)/(25)`

`sin \beta = (4)/(5)`

To find
`cos(\alpha - \beta)`.

As per Formula:

`cos(A-B) =cos A cosB+ sinA sinB`

Here, A is
`\alpha` and B is
`\beta`.

So, formula becomes

`cos(\alpha -\beta)=cos\apha cos\beta+sin\alpha sin \beta ..... (1)`

Using the following identity to calculate
`sin\alpha \text{ and } cos \beta`:

`sin^2 \theta + cos^2 \theta = 1`

`sin^(2)\alpha+\frac {7^(2) }{25^(2) } = 1\\ \Rightarrow sin\alpha = \sqrt{1-(49)/(625)}\\\Rightarrow sin\alpha = (24)/(25)`

Similarly,

`\frac {4^(2) }{5^(2) } + cos^(2)\beta = 1\\ \Rightarrow cos\beta = \sqrt{1-(16)/(25)}\\\Rightarrow cos\beta = (3)/(5)`

Putting values in equation (1):

`cos(\alpha -\beta ) = (7)/(25) * (3)/(5) + (24)/(25) * (4)/(5)\\\Rightarrow (21+96)/(125)\\\Rightarrow (117)/(125)`