Given any right triangle with an acute angle theta, a) Explain why csc theta is always greater than or equal to 1
b) Explain why cos theta is always less than or equal to 1
start with b)
since cos Ø = adjacent/hypotenuse
and the hypotenuse in any right-angled triangle is of course greater than either of the other sides, the ratio of adjacent/hypotenuse must be a proper fraction, that is, it must be less than 1
a) since the secant is the reciprocal of the cosine, and the cosine is always less than 1, then the secant must always be greater than one.
(if you "flip" a proper fraction, it becomes an improper fraction)
find the percent roung your answer to the nearest whole percent when necessary
$6 tip for a $40 dinner
a) Well, csc(theta) is the abbreviation for the cosecant of theta. In a right triangle, the cosecant of an acute angle theta is defined as the ratio of the hypotenuse (let's call it h) to the length of the side adjacent to theta (let's call it a).
Now, in any right triangle, the longest side is always the hypotenuse. Therefore, h is always greater than or equal to a. And since csc(theta) = h/a, we can conclude that csc(theta) is always greater than or equal to 1. But hey, don't get hypotenuse about it!
b) Oh, cosine! Let me tell you why cos(theta) is always less than or equal to 1. Cosine is defined as the ratio of the length of the side adjacent to theta (let's call it a) to the hypotenuse (let's call it h).
Now, since a is always shorter than h in a right triangle, it means that a/h is always less than or equal to 1. So, cos(theta) is always less than or equal to 1. It really knows how to keep things down to earth, unlike my attempts at comedy!
a) The cosecant function, csc(theta), is defined as the reciprocal of the sine function, sin(theta). In a right triangle, the sine of an acute angle, such as theta, is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Since the hypotenuse is always the longest side of the right triangle, the length of the side opposite theta (let's call it "a") will be less than or equal to the length of the hypotenuse (let's call it "c").
Therefore, sin(theta) = a/c, and the reciprocal of sin(theta), which is csc(theta), is equal to c/a. Since a is less than or equal to c, c/a is always greater than or equal to 1. Hence, csc(theta) is always greater than or equal to 1.
b) The cosine function, cos(theta), is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse in a right triangle. In a right triangle, the hypotenuse is always the longest side, while the side adjacent to an acute angle, such as theta, is always shorter than or equal to the hypotenuse.
Let's consider a scenario where the side adjacent to theta (let's call it "b") is equal to the hypotenuse (let's call it "c"). In this case, cos(theta) = b/c = c/c = 1, as the numerator and denominator cancel out.
Now, if we consider a scenario where b is shorter than c, then cos(theta) = b/c is less than 1 because the numerator is smaller than the denominator.
In either case, cos(theta) is always less than or equal to 1 because the adjacent side is either equal to or shorter than the hypotenuse. Thus, cos(theta) is always less than or equal to 1.
a) The cosecant function (csc) is defined as the reciprocal of the sine function (sin). In a right triangle, the sine of an acute angle is equal to the length of the opposite side divided by the length of the hypotenuse. Since the sine function is always between -1 and 1, the reciprocal, csc theta, will always be greater than or equal to 1. This is because dividing a number by a value between -1 and 1 will yield a larger number.
To visualize this, consider a right triangle with an acute angle theta. The side opposite to theta is always shorter than or equal to the hypotenuse. When you take the reciprocal by dividing 1 by a number smaller than or equal to the hypotenuse, you get a value greater than or equal to 1.
b) The cosine function (cos) is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In a right triangle, the length of the adjacent side is always equal to or smaller than the hypotenuse. Since the cosine function calculates this ratio, cos theta will always be less than or equal to 1.
To understand why cos theta is always less than or equal to 1, visualize a right triangle with an acute angle theta. The adjacent side to theta is always shorter than or equal to the hypotenuse. When you divide the length of the adjacent side by the length of the hypotenuse, you get a value between 0 and 1. This means that the cosine of any acute angle theta is always less than or equal to 1.