Answer:
sin(x°) ≡ cos(y°) = 3/5
Step-by-step explanation:
The definitions of the trig functions can answer the question about the relationship of sine and cosine. The Pythagorean theorem can help find the values of the trig functions.
Hypotenuse
The hypotenuse of this right triangle is found using the Pythagorean theorem.
a² +b² = c² . . . . . . sum of squares of sides is square of hypotenuse
3² +4² = c²
9 +16 = 25 = c²
c = √25 = 5
The length of hypotenuse OP is 5 units.
Trig functions
The mnemonic SOH CAH TOA reminds you of the relationships between sides of a right triangle and trig functions of the acute angles:
Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Values
sin(x°) = Opposite/Hypotenuse = 3/5
cos(y)° = Adjacent/Hypotenuse = 3/5
Relationship
The side opposite angle x° is the side adjacent to angle y°, so the sine of x will always be identical to the cosine of y. (And vice versa.) That is, the sine of an angle is identical to the cosine of its complement.
__
Additional comment
You will notice that the side lengths in this triangle, {3, 4, 5}, are consecutive integers. These are the only consecutive integer side lengths that form a right triangle, and the only lengths that are an arithmetic sequence. The "Pythagorean triple" {3, 4, 5} and its multiples show up many times in algebra, geometry, and trig problems. It is useful to remember.
