Answer:
[tex]\begin{gathered} \sin45=\frac{\sqrt{2}}{2} \\ \cos45=\frac{\sqrt{2}}{2} \\ \tan45=1 \end{gathered}[/tex]Explanation:
Given:
45°-45°-90° triangle.
To find:
The numerical values of the trigonometric ratios.
Let's sketch the given triangle;
Since sides AC and CB are congruent, we can choose the length of their sides to be 1 and go ahead and solve for the length of side AB which is x using the Pythagorean theorem as seen below;
[tex]\begin{gathered} x^2=1^2+1^2 \\ x^2=1+1 \\ x^2=2 \\ x=\sqrt{2} \end{gathered}[/tex]We can now determine the values of each trigonometric ratio as seen below;
[tex]\sin45=\frac{opposite\text{ side to angle 45}}{hypotenuse}=\frac{1}{\sqrt{2}}=\frac{1*\sqrt{2}}{\sqrt{2}*\sqrt{2}}=\frac{\sqrt{2}}{2}[/tex][tex]\cos45=\frac{adjacent\text{ side to angle 45}}{hypotenuse}=\frac{1}{\sqrt{2}}=\frac{1*\sqrt{2}}{\sqrt{2}*\sqrt{2}}=\frac{\sqrt{2}}{2}[/tex][tex]\tan45=\frac{opposite\text{ side to angle 45}}{adjacent\text{ side to angle 45}}=\frac{1}{1}=1[/tex]