Answer:
The ratios of sin(x) and cos(y) are equal.
sin(x) = ³/₅
cos(y) = ³/₅
Step-by-step explanation:
Trigonometric ratios
[tex]\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}[/tex]
where:
- [tex]\theta[/tex] 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 triangle:
- [tex]\theta[/tex] = x
- O = 3
- H = OP
Substitute the given values into the sine ratio to find sin(x):
[tex]\implies \sf \sin x=\dfrac{3}{OP}[/tex]
From inspection of the given triangle:
- [tex]\theta[/tex] = y
- A = 3
- H = OP
Substitute the given values into the cosine ratio to find cos(y):
[tex]\implies \sf \cos y=\dfrac{3}{OP}[/tex]
Therefore, the ratios of sin(x) and cos(y) are equal in a right triangle where the angles x and y are the non-right angles.
To find the actual values of sin(x) and cos(y), find the measure of the hypotenuse (OP) by using Pythagoras Theorem:
[tex]\implies \sf a^2+b^2=c^2[/tex]
[tex]\implies \sf 3^2+4^2=OP^2[/tex]
[tex]\implies \sf OP=\sqrt{3^2+4^2}[/tex]
[tex]\implies \sf OP=5[/tex]
Substitute the found value of the hypotenuse (OP) into the found ratios.
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