Question

Why are the sime and cosine values of an acute angle in a right triangle always less than 1?

Answer 1

Sine and cosine values are derived from the unit circle, which is a circle with a radius of 1, where the midpoint of the circle is at (0, 0) of the y and x-axis.

Draw a horizontal line from the midpoint (0, 0) to (1, 0) on the x-axis, and that shall be your initial line when the angle equals to 0°.

The terminal line is the line which stretches from (0, 0) to any point on the circumference of the circle.

The angle between the initial line and the terminal line will only be acute if the terminal line is in the position region, where x and y values are positive. This is called the first quadrant of the unit circle.

Recall the basic formula for Sine and Cosine:
SOH CAH, and let x be an angle in a right-angled triangle.
sin(x) = (the side Opposite the angle) / (Hypotenuse of triangle)
cos(x) = (the side Adjacent to the angle) / (Hypotenuse of the triangle)

Back to the unit circle, if you were to draw a line from the tip of the terminal line (the tip touches the circumference of circle) to the initial line, where it meets the initial line perpendically, you'll see a right-angled triangle.

sine of this angle between the terminal line and the initial line will then be:
sin(x) = (the side Opposite the angle) / (Hypotenuse)

{Since the circle is a unit circle, where radius is equals to 1, and since the hypotenuse of the right-angled triangle is the terminal line and is also the radius of the circle, the hypotenuse equals to 1}
sin(x) = (Opposite) / 1
= Opposite
Thus, sin(x) is actually the y-value of the tip of terminal line (the tip touches the circumference of the unit circle). When the angle between terminal and initial lines is 90°, the tip of the terminal line will touch the circumference of the unit circle at (0, 1), which is the highest point of the unit circle. Since the highest y-value of the unit circle is 1 when the angle is 90°, if the angle is an acute angle (I.e. an angle less than 90°), the y-value of the tip of terminal line will always be less than 1.

To simplify it:
When angle x is 0° < x < 90°, sin(x) = Opposite = y-value which is < 1

For cos(x),
cos(x) = (Adjacent) / (Hypotenuse)

Similarly, the hypotenuse is equal to the length of the terminal line, and also equals to the radius of the unit circle (I.e. 1).
cos(x) = (Adjacent) / 1
= Adjacent
The adjacent side of the right-angled triangle is the x-value of the unit circle. The length of adjacent side is from (0, 0) to the perpendicular line drawn from the terminal line down to the initial line.

So, when the angle is equals to 0°, cos(0°) = 1, which is the length of the initial line from (0, 0) to (1, 0). If the angle is an acute angle, the length of initial line or adjacent side will be less than 1.

To simplify:
When angle x is 0° < x < 90°, cos(x) = Adjacent = x-value, which is < 1

Hope this helps! :) I understand that this is a little lengthy as compared to most answers on Brainly, and that this may be complicated or confusing to whoever who reads it the first time, so feel free to ask me more questions to clarify your doubts if you have :)