Why do we need angle values greater than 90 degrees? (2024)

Hemant said:

I can't understand how can we assign values greater than 90 to trigonometric functions as right angle triangle can't exist if one angle is more than 90 degree. For example if I say sin 30 according to me it means that ratio of perpendicular and hypotenuse is 1/2 at 30 degree but how can we say something like this in angles greater than 120.

It's called a generalisation. Imagine the unit circle and start with your right-angle triangle in the first quadrant. You notice that:

##x = \cos \theta \ ## and ##y = \sin \theta##

As you continue round the circle, you could extend your definition of sine and cosine by taking these equations to define ##\sin \theta## and ##\cos \theta##.

And then you have something even more useful than restricting yourself to angles less than ##\pi/2##.

Hemant said:

I can't understand how can we assign values greater than 90 to trigonometric functions as right angle triangle can't exist if one angle is more than 90 degree.

Have you studied trigonometry as it is defined using the unit circle? If so, you understand how it is done. Perhaps your question is why it is done. Do you want to know why defining the trigonometric functions for all angles is useful?

Stephen Tashi said:

Have you studied trigonometry as it is defined using the unit circle? If so, you understand how it is done. Perhaps your question is why it is done. Do you want to know why defining the trigonometric functions for all angles is useful?

Yes please.

Perhaps @Frigus noticed that ##\tan \frac{\pi}{2} = \tan 90^{\text{o}}## still remains undefined, even when generalizing trig functions with a unit circle. The same problem we have with a right triangle with two ##90^{\text{o}}## angles we have with unit-circle-based definition of ##\text{tangent}##. A "right triangle" with two ##90^{o}## cannot exist. How can there be "right triangles" with obtuse angles?

Agent Smith said:

Yes please.

Perhaps @Frigus noticed that ##\tan \frac{\pi}{2} = \tan 90^{\text{o}}## still remains undefined, even when generalizing trig functions with a unit circle. The same problem we have with a right triangle with two ##90^{\text{o}}## angles we have with unit-circle-based definition of ##\text{tangent}##. A "right triangle" with two ##90^{o}## cannot exist. How can there be "right triangles" with obtuse angles?

The aim of extending or generalizing right triangle trig is not to provide definitions for such expressions as ##\tan(\pi/2)## or ##\csc(0)##. It is to be able to define values for the six trig functions for all real angle values, not just those between 0 and 90°. Of course, the trig functions that are defined in terms of division (tangent, cotangent, secant, cosecant) have domains that don't permit certain values.

Regarding your comment about the impossibility of a right triangle with two right angles, that's true if we're talking only about plane surfaces. However, without this limitation it's possible to have a right triangle with three right angles. Suppose you're standing at the north pole. You walk due south for one mile, and then turn left, making a 90° angle. Head due east for one mile and turn left again, making another 90° angle. Head due north for one mile to reach your starting point at the north pole. Your path determines an equilateral triangle all of whose angles are 90°.

@Mark44 , muchas gracias.

##\sin (\theta) = \sin (180^o - \theta)##

##\cos (\theta) = \cos (360^o - \theta)##

##\tan (\theta) = \tan (180^o + \theta)##

Would I be correct to say that the trig function values for obtuse and reflex angles are equal to the trig function values of their corresponding acute angles.

Agent Smith said:

Would I be correct to say that the trig function values for obtuse and reflex angles are equal to the trig function values of their corresponding acute angles.

This is a bit too general. You can answer your own question by using the trig identities for sums and differences of angles. From them you should be able to see that the sine of an angle and its supplement are equal, but the cosine of an angle and its supplement differ in sign. IOW ##\cos(\theta) = -\cos(\pi - \theta)##.

Mark44 said:

You can answer your own question

I wish!

I was only trying to give the OP an idea of what I felt was some kind of pair-matching between reflex and obtuse angles and their corresponding acute angles (supplementary i.e. sum to 180 degrees and sum-to-360 degrees) in re their trig function values.

Mark44 said:

You can answer your own question by using the trig identities for sums and differences of angles.

Agent Smith said:

I wish!

Why not? The sum and difference trig identities are in every textbook on trig or can easily be found online, like on wikipedia.

Agent Smith said:

I was only trying to give the OP an idea of what I felt was some kind of pair-matching between reflex and obtuse angles and their corresponding acute angles (supplementary i.e. sum to 180 degrees and sum-to-360 degrees) in re their trig function values.

I'm not sure the OP is still paying attention in a thread that is almost five years old and who hasn't been heard of for more than a year.