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

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In summary: If so, you may need to study some applications of trigonometry in physics, engineering, and other fields. In summary, the conversation discusses assigning values greater than 90 to trigonometric functions and how this is possible even when dealing with angles greater than 120. This is because trigonometric functions can be generalized to the full circle, rather than just right triangles, and are useful for modeling repetitive phenomena in various fields.

  • #1

Frigus

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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.

  • #2

fresh_42

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This is because we do not consider only (right) triangles, but the full circle instead which we divide into degrees. E.g. look at a compass and how pilots and captains measure their direction. And even in triangles, there are triangles with angles greater than 90° or 120°, and we also consider the outer angles, the complementary angles to the inner ones.

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PeroK

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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##.

  • #4

WWGD

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You can even define a full coordinate system, polar coordinates, using sin, cos.

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HallsofIvy

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IF you are only using the "trig functions" on right triangles then there is no reason to use angles greater than 90 degrees. But generalizations of the trig functions (sometimes renamed "circular functions") are very useful as "periodic functions" modeling repetitive phenomena. As functions, we want them defined for all real numbers.

  • #6

Stephen Tashi

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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?

  • #7

Agent Smith

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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?

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Mark44

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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°.

  • #9

Agent Smith

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@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.

  • #10

Mark44

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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)##.

  • #11

Agent Smith

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Mark44 said:

You can answer your own question

I wish! Why do we need angle values greater than 90 degrees? (1)

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.

  • #12

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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 do we need angle values greater than 90 degrees? (2)

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.

  • #13

Agent Smith

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@Mark44 I find it intriguing that generalizing trig functions with a unit circle gave meaning to trig functions for all angles (not just acute angles) except ##\tan \frac{\pi}{2}##.
##\cos 0 = 1## (a degenerate triangle T1)
##\sin \frac{\pi}{2} = 1## (another degenerate triangle T2)
##\tan \frac{\pi}{2} = \text{undefined}## (the same degenerate triangle T2)

Related to Why do we need angle values greater than 90 degrees?

1. Why can't we just use angles less than 90 degrees?

Angles greater than 90 degrees are necessary in certain situations because they allow us to measure and describe angles that are larger than a right angle. For example, in geometry and trigonometry, angles greater than 90 degrees are used to calculate and describe obtuse and reflex angles.

2. What real-life applications require angles greater than 90 degrees?

Angles greater than 90 degrees are commonly used in navigation, engineering, and architecture. In navigation, angles greater than 90 degrees are used to measure bearings and headings. In engineering and architecture, angles greater than 90 degrees are used to design structures and determine the direction and force of forces acting on them.

3. Can't we just use negative angles instead of angles greater than 90 degrees?

Negative angles are used to describe angles that are less than 0 degrees, but they cannot accurately represent angles greater than 90 degrees. Negative angles are also not commonly used in real-life applications, making it more practical to use angles greater than 90 degrees.

4. How do angles greater than 90 degrees relate to circles and radians?

In a circle, there are 360 degrees or 2π radians. Angles greater than 90 degrees can be converted to radians by multiplying the degree measure by π/180. For example, an angle of 180 degrees is equal to π radians, and an angle of 270 degrees is equal to 3π/2 radians.

5. Are there any special properties or rules for angles greater than 90 degrees?

Angles greater than 90 degrees follow the same basic properties and rules as angles less than 90 degrees. However, in some cases, they may require different formulas or calculations. For example, the sine, cosine, and tangent functions are defined differently for angles greater than 90 degrees.

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

                      FAQs

                      What if an angle is greater than 90 degrees? ›

                      Acute angles measure less than 90 degrees. Right angles measure 90 degrees. Obtuse angles measure more than 90 degrees.

                      Why are 90 degree angles important? ›

                      Equal Measures: At the junction of two perpendicular lines, there are four equal angles, each measuring ninety degrees. This equivalence offers a fundamental concept in geometry and trigonometry and makes computations and geometric proofs easier.

                      What is an angle measures greater than 90? ›

                      An obtuse angle measures more than 90 degrees (a right angle) but less than 180 degrees (a straight angle). Thus, the obtuse angle degree range is (90°,180°).

                      Can angle of incidence be greater than 90 degrees? ›

                      So in any case, the angle of incidence and the refraction cannot exceed 90 degrees, since if you exceed 90 degrees, the incident light ray will be actually angle of refraction and vice versa. So in terminology, you cannot have an angle of more than 90 degrees.

                      What does an angle other than 90 degrees mean? ›

                      Acute Angle - An angle less than 90 degrees. Right Angle - An angle that is exactly 90 degrees. Obtuse Angle - An angle more than 90 degrees and less than 180 degrees. Straight Angle - An angle that is exactly 180 degrees. Reflex Angle - An angle greater than 180 degrees and less than 360 degrees.

                      What if each angle is less than 90 degree? ›

                      A triangle in which each angle is less than 90 degrees is called an acute angled triangle. For example, the triangle shown below is an acute angled triangle. Q. A triangle in which each angle is less than 90 degrees is called ___.

                      Why are angles important in real life? ›

                      Angles are all around us. Position, direction, precision, and optimization are some ways in which people use angles in their daily life. Carpenters use them to measure precisely to build doors, chairs, tables, etc. Athletes use them to gauge the distances of a throw and to enhance their performance in sports.

                      Why is the critical angle 90 degrees? ›

                      The critical angle is the angle of incidence where the angle of refraction is 90°. The light must travel from an optically more dense medium to an optically less dense medium. Figure 5.15: When the angle of incidence is equal to the critical angle, the angle of refraction is equal to 90°.

                      What is the rule for a 90-degree angle? ›

                      Here are the rotation rules: 90° clockwise rotation: (x,y) becomes (y,−x) 90° counterclockwise rotation: (x,y) becomes (−y,x)

                      What is the law of cosines for angles greater than 90? ›

                      If C = 90 degrees, 2ab(cos(90)) = 0 and we get the old Pythagorean Theorem. If C < 90 degrees, then cos(C) is positive, and we wind up adding to c^2 to make the equation balance. If C > 90 degrees, then cos(C) is negative, and we wind up subtracting from c^2 to make the equation balance.

                      Can sin be greater than 90 degrees? ›

                      Angles greater than 90 degrees can have sines as follows: there is a symmetry between angles on either side of 90 degrees, so angles of [90 + d] degrees have the same sine as angles of [90 - d] degrees. For example, the sine of 110 degrees = sine [90 - 20] degrees = sine 70 degrees.

                      What if any one angle is greater than 90 degree? ›

                      An obtuse-angled triangle is a triangle in which one of the interior angles measures more than 90° degrees.

                      What happens when an angle is greater than a critical angle? ›

                      When the angle of incidence is greater than the critical angle none of it is refracted, the ray is totally internally reflected, and the law of reflection is obeyed, i = r.

                      Why does light not bend at 90 degrees? ›

                      A beam of light bends when it enters glass at an angle. This is due to refraction of light. It does not bend if it enters the glass at right angles because no refraction will occur in this case, the angle of incidence in this case is zero and angle of refraction is also zero.

                      What is an angle greater than 90 degrees triangle? ›

                      An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.

                      Can two angles add up to be greater than 90 degrees? ›

                      If the sum of two angles is 180 degrees then they are said to be supplementary angles, which form a linear angle together. Whereas if the sum of two angles is 90 degrees, then they are said to be complementary angles, and they form a right angle together.

                      What is an angle greater than 90 degrees and no equal sides? ›

                      Answer: The correct answer is obtuse scalene. A scalene triangle is one with three unique side lengths and angle measures. It is obtuse because the largest angle is greater than 90°.

                      What is an angle whose measure is greater than 90 degrees Cannot be? ›

                      An angle whose measure is greater than 90°, but less than 180° is called obtuse angle.

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