Question

In a right triangle with acute angle of measure theta, sintheta=7 /9. What is the value of costhetatheta? Draw a diagram as part of your response.

Answer 1

The value of
`cos\:\theta` is
`(4√(2))/(9)`

Explanation:

The Pythagorean Theorem,
`a^2+b^2=c^2` can be used to find the length of any side of a right triangle.

We know that
`sin\:\theta=(7)/(9)` this means that the ratio of the opposite side to the hypotenuse is
`(7)/(9)`. So
`a=7` and
`c=9`.

To find adjacent side (b) we apply the Pythagorean Theorem as follows:

`a^2+b^2=c^2\\(7)^2+b^2=(9)^2\\b^2=(9)^2-(7)^2\\b^2=81-49\\b=√(32) \\b=4√(2)`

We need to use the fact that
`cos\:\theta=(adjacent)/(hypotenuse)`

We have the adjacent side =
`4√(2)` and the hypotenuse = 9, so

`cos\:\theta=(4√(2))/(9)`