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:The value of [tex]cos\:\theta[/tex] is [tex]\frac{4\sqrt{2}}{9}[/tex]Step-by-step explanation:The Pythagorean Theorem, [tex]a^2+b^2=c^2[/tex] can be used to find the length of any side of a right triangle.We know that [tex]sin\:\theta=\frac{7}{9}[/tex] this means that the ratio of the opposite side to the hypotenuse is [tex]\frac{7}{9}[/tex]. So [tex]a=7[/tex] and [tex]c=9[/tex].To find adjacent side (b) we apply the Pythagorean Theorem as follows:[tex]a^2+b^2=c^2\\(7)^2+b^2=(9)^2\\b^2=(9)^2-(7)^2\\b^2=81-49\\b=\sqrt{32} \\b=4\sqrt{2}[/tex]We need to use the fact that [tex]cos\:\theta=\frac{adjacent}{hypotenuse}[/tex]We have the adjacent side = [tex]4\sqrt{2}[/tex] and the hypotenuse = 9, so[tex]cos\:\theta=\frac{4\sqrt{2}}{9}[/tex]