1. We are given a right triangle with vertices D (top-left), F (bottom-left, right angle), and E (bottom-right). Angles are labeled \(\alpha\) at D and \(\beta\) at E. 2. The sides are labeled as follows: - \(|DF|\) is the side opposite \(\beta\) and adjacent to \(\alpha\). - \(|DE|\) is the hypotenuse. - \(|EF|\) is the side opposite \(\alpha\) and adjacent to \(\beta\). 3. Recall the definitions of sine, cosine, and tangent for an angle in a right triangle: - \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\) - \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\) - \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) 4. Using these definitions, we fill in the missing goniometric values: - \(\beta = \frac{|DF|}{|DE|} = \sin(\beta)\) because \(|DF|\) is opposite \(\beta\) and \(|DE|\) is hypotenuse. - \(\beta = \frac{|DF|}{|EF|} = \tan(\beta)\) because \(|DF|\) is opposite \(\beta\) and \(|EF|\) is adjacent. - \(\alpha = \frac{|DF|}{|DE|} = \cos(\alpha)\) because \(|DF|\) is adjacent to \(\alpha\) and \(|DE|\) is hypotenuse. - \(\alpha = \frac{|EF|}{|DF|} = \tan(\alpha)\) because \(|EF|\) is opposite \(\alpha\) and \(|DF|\) is adjacent. - \(\alpha = \frac{|EF|}{|DE|} = \sin(\alpha)\) because \(|EF|\) is opposite \(\alpha\) and \(|DE|\) is hypotenuse. - \(\beta = \frac{|EF|}{|DE|} = \cos(\beta)\) because \(|EF|\) is adjacent to \(\beta\) and \(|DE|\) is hypotenuse. Final answers: \(\beta = \sin(\beta) = \frac{|DF|}{|DE|}\) \(\beta = \tan(\beta) = \frac{|DF|}{|EF|}\) \(\alpha = \cos(\alpha) = \frac{|DF|}{|DE|}\) \(\alpha = \tan(\alpha) = \frac{|EF|}{|DF|}\) \(\alpha = \sin(\alpha) = \frac{|EF|}{|DE|}\) \(\beta = \cos(\beta) = \frac{|EF|}{|DE|}\)