Solving expressions using 45-45-90 special right triangles

Special right triangles 45 45 90

Special triangles are a way to get exact values for trigonometric equations. Most trig questions you’ve done up till now have required that you round answers in the end. When numbers are rounded, it means that your answer isn’t exact, and that’s something that mathematicians do not like. Special triangles take those long numbers that require rounding and come up with exact ratio answers for them.

There’s not a lot of angles that give clean and neat trigonometric values. But for the ones that do, you will have to memorize their angles’ values in tests and exams. These are the ones you’ll most typically use in math problems as well. For a list of all the different special triangles you will encounter in math.

One of these triangles is the 45 45 90 triangle. It is an isosceles triangle, with two equal sides. Since you’ll also find that this triangle is a right-angled triangle, we know that the third side that is not equal with the others is the hypotenuse. You also happen to know a nice formula to figure out what the length of the hypotenuse is (the Pythagorean Theorem) and we’ll show you how it will be used. If you wanted to take a look at more examples of the 45 45 90 triangle, take a look at this interactive online reference for this special right triangle.

How to solve 45 45 90 triangle

To help demonstrate what the special right triangle with 45 45 90 as its angles looks like, as well as explain the values that you’ll have to work with going forward, we’ll use the below example. It shows a standard 45 45 90 triangle that can help you understand the ratios that come about when this triangle is used.

1 is chosen to be used as the length of the sides that are equal in this special triangle, since that’s the simplest to work with. So how do we find the hypotenuse?

45-45-90 triangle
a 45 45 90 special triangle triangle

This is an isosceles right triangle. Since it is a right triangle, we can use Pythagorean Theorem to find the hypotenuse.

c2=a2+b2{c^2} = {a^2} + {b^2}

c2=12+12{c^2} = {1^2} + {1^2}

c2=1+1=2{c^2} = 1 + 1 = 2

c=2c = \sqrt 2

With the hypotenuse, we have information to determine the following:

sin45=12sin\;45^\circ = \;\frac{1}{{\sqrt 2 }}

cos45=12cos\;45^\circ = \frac{1}{{\sqrt 2 }}\;

tan45=11=1tan\;45^\circ = \frac{1}{1} = 1\;

You can see that we are looking at the “theta” of 45 degrees, and you should remember SOHCAHTOA, which helps you remember which sides you’ll need to take to find the sine, cosine, and tangent. That is how we got that the sine is 1/sqrt(2), since 1 is the length of the side that is opposite of 45 degrees, and the hypotenuse is sqrt(2). For cosine, you’ll need the adjacent over the hypotenuse, which gives you 1/sqrt(2). Lastly, for the tangent, it’s opposite over adjacent, giving you 1/1, or in a more simplified form, just 1.

Knowing that you’ll need to memorize these values, you can opt to commit them to memory, or you can redraw this triangle and use SOHCAHTOA to help you find the angle’s ratios. Either way, we hope that by explaining the components of the triangle to you, you now have a better understanding of the 45 45 90 special triangle and how its ratios came about.

Solving expressions using 45-45-90 special right triangles


sin45=12 \sin 45^\circ = {1 \over \sqrt 2}
cos45=12 \cos 45^\circ = {1 \over \sqrt 2}
tan45=11=1 \tan 45^\circ = {1 \over 1} = 1

45-45-90 special right triangle
  • 1.
    What are special triangles?

  • 2.
    If θ=45 \theta = 45^\circ , find the exact value of following expressions.
    3sin2θ3 \sin^2 \theta

    tan(3θ75)\tan (3 \theta - 75^\circ)

    cos2(1202θ)\cos^2(120^\circ - 2 \theta)