[G.SRT.8.1] Similarity, Right Triangles, and Trigonometry #8.1

Objective

Common Core Text:

• • [G.SRT.8.1] Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90°and 45°, 45°, 90°). (CA)

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Explanation

There are two kinds of special right triangles that come up more than others.

30°-60°-90°

The first is called a 30°-60°-90° triangle. It is important because it is half of an equilateral triangle. Lets look at an explanation.

First, start with an equilateral triangle. In an equilateral triangle, all angles and sides are the same. For angles, that means all angles must be 60° (because 60+60+60 = 180). For sides, they could be anything, so we'll just call the length s

Cut it in half

What's left over?

Now we can finally find our trigonometric ratios.

Sine of an Angle =

sin30° = = = 0.5

Cosine of an Angle = cos30° =

= = 0.866

sin60° = = = 0.866

cos60° = = = 0.5

Remember that sina = cosb if a and b are complementary angles.

45°-45°-90°

The second special triangle is called a 45°-45°-90° triangle. It is important because it is the only possible isosceles right triangle. Lets look at an explanation.

First, start with an isosceles right triangle.

Let the sides be length s. We need to find the hypotenuse. Using the pythagorean theorem, we get

• a² + b² = c²

• s² + s² = c²

  • 2s² = c²

• c = s√2

We can use these sides to find the trigonometric ratios of the triangle

= =

Summary

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