With reference to the right triangle,, explain why the expressions y = sin x and y = cos ((pi/2) - (x)) give the same results for all the values of x. Discuss whether or not this same relationship exists for angles greater than (pi/2).
To understand why the expressions y = sin x and y = cos ((π/2) - x) give the same results for all values of x in a right triangle, we need to examine the definitions and properties of sin and cos functions.
First, let's clarify the terms used:
- Right Triangle: A triangle that has one angle measuring 90 degrees (π/2 radians).
- Sin (Sine) Function: In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- Cos (Cosine) Function: In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
Now, let's analyze the expressions y = sin x and y = cos ((π/2) - x):
1. y = sin x:
The sine function gives the ratio of the length of the side opposite the angle (y in this case) to the length of the hypotenuse. This means that for any angle x in a right triangle, the value of y will depend on the length of the side opposite that angle.
2. y = cos ((π/2) - x):
The cosine function gives the ratio of the length of the adjacent side (y in this case) to the length of the hypotenuse. By using the identity cos ((π/2) - x) = sin x, we can see that the expression is equivalent to y = sin x. This means that for any angle x in a right triangle, the value of y will still depend on the length of the side opposite that angle.
Therefore, the expressions y = sin x and y = cos ((π/2) - x) will always give the same results for all values of x in a right triangle.
However, it's important to note that this relationship does not hold for angles greater than π/2 (90 degrees). As angles become greater than π/2, the sides of the triangle that were previously opposite and adjacent will switch roles. In other words, the side that was previously adjacent will become the opposite side, and vice versa. As a result, the cosine function will measure the length of the side that was previously opposite the angle, and the sine function will measure the length of the side that was previously adjacent.
In conclusion, the expressions y = sin x and y = cos ((π/2) - x) give the same results for all values of x in a right triangle, but this relationship does not hold for angles greater than π/2.