trigonometry?

Question

how do you know from the definition of sine, that in a right triangle , the sine of an acute angle is always less then 1? How do you know that the cosine of an acute angle is always less than 1?

Answer 1

In a unit circle, the radius (hypotenuse) is 1 so the y (vertical) or the x (horizontal) can never be more than 1. If the y is 1 (the most it can be) then you have sin(θ) = y / r
so sin(θ) = 1 /1
arcsin(1) = 90° or π / 2.
If the x is 1 (the most it can be) then you have cos(θ) = x / r
so cos(θ) = 1 /1
arccos(1) = 0°
Therefore the sine and cosine of an acute angle is always less than 1.

Answer 2

The sine of any angle can never be greater than 1, no matter what the angle is. The same is true for cosine. So by the definition of sine and cosine, they are always less than or equal to 1.

The only time the sine is equal to 1 is when the angle is 90 degrees, which is not acute. The only time the cosine is equal to 1 is when the angle is 0 degrees, which is also not an acute angle.

Thereore all acute angles have sine less than 1.

Answer 3

Because at the sine 90 degrees a triangle becomes a straight line equal to 1, likewise the cosine at 0 degrees.

Answer 4

the sine of an angle is the ratio of the opposite side to the hypotenuse.
If one angle of a right triangle is right, the others must be acute.
the hypotenuse is the longest side in a right triangle since it is the side opposite the largest angle
hence opposite/hypotenuse must be less than one. same goes for cosine, with opposite replaced by adjacent.
Note; I tried not using the unit circle in my argument, since you asked by looking at the right triangle.

Answer 5

Trigonometry is a branch of arithmetic that learn triangles and the relationships between their components and the angles between those components. Trigonometry defines the trigonometric purposes, which describe those relationships and function applicability to cyclical phenomena, alongside with waves. the sphere developed for the period of the 0.33 century BC as a branch of geometry used significantly for astronomical learn.[2] it is likewise the beginning place of the clever artwork of surveying.

Answer 6

You know that a hypotenuse is the longest side of a right angled triangle.

You also know that ....

Sin theta = Opposite side / Hypotenuse

Cos theta= Adjacent side/ Hypotenuse

As hypotenuse is longer than both 'opposite side' and 'adjacent side' both sin theta and cos theta return "less than 1" values. (Reason is hypotenuse (denominator) is always greater than respective numerators and so we get less than one values alone!


Regards!

Answer 7

The hypotenuse is the longest side = r, say.
Let other sides be x (horizontal) and y (vertical)
let angle be Ø.
sin Ø = y / r
Now y < r, thus sin Ø < 1
cos Ø = x / r
Now x < r, thus cos Ø <1