Answer:
[tex]x =6[/tex]
[tex]y=6\cdot \sqrt{2}[/tex]
Step-by-step explanation:
Trigonometric Ratios
The ratios of the sides of a right triangle are called trigonometric ratios. There are six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent.
The longest side of the right triangle is called the hypotenuse and the other two sides are the legs.
Choosing any of the acute angles as a reference, it has an adjacent side and an opposite side. The trigonometric ratios are defined upon those sides.
The image shows a right triangle where the angle of 45° has x as the opposite leg, 6 as the adjacent leg, and y as the hypotenuse. The trigonometric ratio that applies here is the cosine ratio, defined as:
[tex]\displaystyle \cos\theta=\frac{\text{adjacent leg}}{\text{hypotenuse}}[/tex]
[tex]\displaystyle \cos 45^\circ=\frac{6}{y}[/tex]
Solving for y:
[tex]\displaystyle y=\frac{6}{\cos 45^\circ}[/tex]
[tex]\cos 45^\circ=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}[/tex]
Substituting:
[tex]\displaystyle y=\frac{6}{\frac{1}{\sqrt{2}}}[/tex]
[tex]y=6\cdot \sqrt{2}[/tex]
Now use the tangent ratio:
[tex]\displaystyle \tan\theta=\frac{\text{opposite leg}}{\text{adjacent leg}}[/tex]
[tex]\displaystyle \tan 45^\circ=\frac{x}{6}[/tex]
Solving for x:
[tex]x=6\cdot\tan 45^\circ[/tex]
[tex]\tan 45^\circ=1[/tex]
Substituting:
[tex]x=6\cdot 1[/tex]
[tex]x =6[/tex]
Answer:
[tex]x =6[/tex]
[tex]y=6\cdot \sqrt{2}[/tex]
