Calculating Angles for a 5-12-13 Triangle

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• 0:04 Pythagorean Triple Problem
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Instructor
Sharon Linde

Sharon has a Masters of Science in Mathematics

Expert Contributor
Alfred Mulzet

Dr. Alfred Kenric Mulzet received his Ph.D. in Applied Mathematics from Virginia Tech. He currently teaches at Florida State College in Jacksonville.

How do you calculate the angles of a triangle when you only know the lengths of the sides? This lesson will go over a step-by-step process to calculate the angles, using a triangle whose sides are 5, 12 and 13 units long.

Pythagorean Triple Problem

Carpenters building structures on and around angled roofs often have to deal with measuring angles. Have you ever had to calculate an angle based on the geometry of a situation? Let's take a look at how to do this for a triangle that measures 5 feet by 12 feet by 13 feet.

The first thing to recognize about this problem is that 5-12-13 is a Pythagorean triple. Pythagoras was a mathematician in ancient Greece that's credited with proving that the squares of the legs of a right triangle when added together are equal to the square of the hypotenuse. You may know it as the familiar a^2 + b^2= c^2 rule learned in algebra classes all over the world.

A Pythagorean triple is simply a set of three integers that are solutions for the Pythagorean theorem. The best known triple is 3-4-5, with 5-12-13 being the next most recognized. Any triangle composed of sides of lengths that match the Pythagorean triple will be a right triangle. That means our triangle has a 90 degree angle for angle C.

Now that we know this is a right triangle, we can use the law of sines, which can be used to relate the length of sides to their opposite angles in triangles. Essentially, it says that the sine of an angle is proportional to the length of the opposite side in any given triangle. Since we know the lengths of three sides plus one of the angles, we can use this law to solve for the missing angles.

For our triangle we now know that a = 5, b = 12, c = 13 and C = 90 degrees. We can then compute that sin C = 1. Substituting all of this into our law of sines equation, we get the following relationships.

Rearranging these equations to solve for sin A and sin B, we get:

We're getting close, but we're not quite there yet. The sine function takes a known angle and gives us a specific value. What we need to do here is the opposite: we have a known sine value and we want to get the measurement of the angle. The function we need is the inverse of the sine function: the arcsine.

Using the above step and our new knowledge of what the arcsine function does, we can write the following equations for our unknown angles:

A = arcsine (5/13)

B = arcsine (12/13)

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Right Triangle Trigonometry:

It is possible to determine the measures of the angles of a right triangle by direct application of the definition of sine, cosine, or tangent. Consider the following triangle:

Now, we wish to determine the value of {eq}\theta. {/eq} Using the identities

{eq}\sin \theta = \displaystyle\frac{\mathrm{opposite}}{\mathrm{hypotenuse}} \\ \cos \theta = \displaystyle\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}} \\ \tan \theta = \displaystyle\frac{\mathrm{opposite}}{\mathrm{adjacent}} {/eq}

we get

{eq}\sin \theta = \displaystyle\frac{8}{17} \\ \cos \theta = \displaystyle\frac{15}{17} \\ \tan \theta = \displaystyle\frac{8}{15} {/eq}

We can use the arc function for any of these equations, and get the exact same answer all three times. So for example,

{eq}\theta = \arcsin \displaystyle\frac{8}{17} \approx 28.1^\circ. {/eq}

Now that we have the value of this angle, the third angle in the triangle must be its complement. Therefore the third angle has a measure of {eq}61.9^\circ. {/eq}

Exercise:

Consider the following triangle:

Find the value of {eq}\theta. {/eq} Now find the value of the third angle.

{eq}\theta = \arcsin \displaystyle\frac35 \approx 36.9^\circ. {/eq}

The third angle is approximately {eq}90 - 36.9^\circ = 53.1^\circ. {/eq}

Exercise 2:

Now use the tangent function with the above triangle to find the value of {eq}\theta. {/eq}

{eq}\theta = \arctan \displaystyle\frac34 \approx 36.9^\circ. {/eq}

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