Answer: [tex]\bold{a)\ sin\ \theta=\dfrac{\sqrt{15}}{4}}[/tex]
[tex]\bold{b)\ cos\ \theta=\dfrac{\sqrt{21}}{5}}[/tex]
[tex]\bold{c)\ sin\ \theta=\dfrac{\sqrt7}{4}}[/tex]
Step-by-step explanation:
The Pythagorean Theorem is: a² + b² = c² where;
- a represents adjacent side
- b represents opposite side
- c represents hypotenuse
[tex]a)\ cos\ \theta=\dfrac{adjacent}{hypotenuse}=\dfrac{1}{4}\\\\1^2+(opposite)^2=4^2\\1 + (opposite)^2=16\\.\ \quad (opposite)^2=15\\.\qquad opposite=\sqrt{15}\\\\sin\ \theta = \dfrac{opposite}{hypotenuse}=\boxed{\dfrac{\sqrt{15}}{4}}[/tex]
[tex]b)\ sin\ \theta=\dfrac{opposite}{hypotenuse}=\dfrac{2}{5}\\\\(adjacent)^2+2^2=5^2\\(adjacent)^2+4=25\\(adjacent)^2\qquad=21\\adjacent\qquad \ =\sqrt{21}\\\\cos\ \theta = \dfrac{adjacent}{hypotenuse}=\boxed{\dfrac{\sqrt{21}}{5}}[/tex]
[tex]a)\ cos\ \theta=\dfrac{adjacent}{hypotenuse}=\dfrac{3}{4}\\\\3^2+(opposite)^2=4^2\\9 + (opposite)^2=16\\.\ \quad (opposite)^2=7\\.\qquad opposite=\sqrt{7}\\\\sin\ \theta = \dfrac{opposite}{hypotenuse}=\boxed{\dfrac{\sqrt{7}}{4}}[/tex]
