Answer:
[tex]\sin(x)=\dfrac{5}{13}[/tex]
[tex]\cos(y)=\dfrac{5}{13}[/tex]
[tex]\sin(x)=\cos(y)[/tex]
Step-by-step explanation:
Pythagoras Theorem explains the relationship between the three sides of a right triangle.
[tex]\boxed{a^2+b^2=c^2}[/tex]
where:
- a and b are the legs of the right triangle.
- c is the hypotenuse (longest side) of the right triangle.
From inspection of the given right triangle:
Substitute the given values into the formula and solve for OP to find the length of the hypotenuse:
[tex]\implies 5^2+12^2=OP^2[/tex]
[tex]\implies 25+144=OP^2[/tex]
[tex]\implies 169=OP^2[/tex]
[tex]\implies OP=\sqrt{169}[/tex]
[tex]\implies OP=13[/tex]
Trigonometric ratios
[tex]\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}[/tex]
where:
- θ is the angle.
- O is the side opposite the angle.
- A is the side adjacent the angle.
- H is the hypotenuse (the side opposite the right angle).
From inspection of the given right triangle, for x°:
Substitute the given values into the sine ratio to find sin(x):
[tex]\implies \sin(x)=\dfrac{5}{13}[/tex]
From inspection of the given right triangle, for y°:
Substitute the given values into the cosine ratio to find cos(y):
[tex]\implies \cos(y)=\dfrac{5}{13}[/tex]
Therefore, sin x° = cos y°.
