Acute Angles Are Less Than 90 Degrees

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In geometry and mathematics, acute angles are angles whose measurements fall between 0 and 90 degrees or has a radian of fewer than 90 degrees. When the term is given to a triangle as in an acute triangle, it means that all angles in the triangle are less than 90 degrees.

It is important to note that the angle must be less than 90 degrees to be defined as an acute angle. If the angle is 90 degrees exactly, though, the angle is known as a right angle, and if it is greater than 90 degrees, it is called an obtuse angle.

The ability of students to identify the different types of angles will greatly help them in finding the measurements of these angles as well as the lengths of the sides of shapes that feature these angles as there are different formulas students can use to figure out missing variables.

Measuring Acute Angles

Once students discover the different types of angles and begin to identify them by sight, it's relatively simple for them to understand the difference between acute and obtuse and be able to point out a right angle when they see one.

Still, despite knowing that all acute angles measure somewhere between 0 and 90 degrees, it may be difficult for some students to find the correct and precise measurement of these angles with the help of protractors. Fortunately, there are a number of tried and true formulas and equations for solving for missing measurements of angles and line segments that make up triangles.

For equilateral triangles, which are a specific type of acute triangles whose angles all have the same measurements, consists of three 60 degree angles and equal length segments on each side of the figure, but for all triangles, the internal measurements of the angles always add up to 180 degrees, so if one angle's measurement is known, it's typically relatively simple to discover the other missing angle measurements.

Using Sine, Cosine, and Tangent to Measure Triangles

If the triangle in question is a right angle, students can use trigonometry in order to find the missing values of the measurements of angles or line segments of the triangle when certain other data points about the figure are known.

The basic trigonometric ratios of sine (sin), cosine (cos), and tangent (tan) relate a triangle's sides to its non-right (acute) angles, which are referred to as theta (θ) in trigonometry. The angle opposite the right angle is called the hypotenuse and the other two sides that form the right angle are known as the legs.

With these labels for the parts of a triangle in mind, the three trigonometric ratios (sin, cos, and tan) can be expressed in the following set of formulas:

cos(θ) = adjacent/hypotenuse
sin(θ) = opposite/hypotenuse
tan(θ) = opposite/adjacent

If we know the measurements of one of these factors in the above set of formulas, we can use the rest to solve for the missing variables, especially with the use of a graphing calculator which has a built-in function for calculating sine, cosine, and tangents.

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Russell, Deb. "Acute Angles Are Less Than 90 Degrees." ThoughtCo, May. 31, 2021, Russell, Deb. (2021, May 31). Acute Angles Are Less Than 90 Degrees. Retrieved from Russell, Deb. "Acute Angles Are Less Than 90 Degrees." ThoughtCo. (accessed June 3, 2023).