Special Triangles and Identities

This chapter explores special triangles and identities. It covers identities and finding exact values of sine, cosine, and tangent of common angles. Before attempting this chapter you must have prior knowledge of basic trigonometry and radians.

tan T = sin T/ cos T

Below is a very simple triangle with hypotenuse 1 unit and other sides labelled x and y.
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…we know according to the above triangle
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….therefore;
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…we also know that;
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…therefore;
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Also;
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…therefore;
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…that concludes that;
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…then;
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Finding trigonometric rations

Here is an example of finding trigonometric ratios. Take that;
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…and find cos θ and tan θ
First lets draw a triangle to help show;
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If;
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…tan;
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…and
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…we can represent this on a triangle as shown below.
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…we can use Pythagoras to find the third side x;
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…simplify and find the values of x.
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Now we can substitute in the value of x on the triangle as shown below;
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We now know that the third side is √8 we can work out cos θ and tan θ we know that;
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…now we also know that;
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… we can check to see where it is true that;
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…we have managed to show that in the above triangle;
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Below are more examples;

Example

If;
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…find sin θ and tan θ and also that;
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…we know that;
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…we draw a triangle showing the known values for the adjacent (1) and hypotenuse (√2) as shown below;
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To find y we can us Pythagoras;
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…so…
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…and…
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…we can check this;
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Example

If;
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…find cos θ and tan θ and also show that;
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We know that;
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We draw a triangle and label it with the known values. Opposite is √3 and hypotenuse is 2 as shown below;
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We can use Pythagoras to find the value of x on the triangle;
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So now we know that;
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…and
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we can check this as shown below;
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sin, cos and tan of 30° and 60°

Below is our first special triangle, the triangle is an equilateral triangle with each side 2 units long and each angle 60°.
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…we can draw a line in the middle to form a perpendicular height as shown below;
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…from that we can form a right angled triangle shown below;
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We can find the height h by using Pythagoras;
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Now we can find exact values of sin, cos and tan for 30° and 60° we know that;
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sin, cos and tan of 45°

The following is another special triangle. The triangle will give you exact values for sine, cosine of 45°
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…we can find the value of h by using Pythagoras
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Now we know that h=√2 so we can label that on the triangle;
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…we now have
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We sometimes write 1/√2 on top, we do this by multiplying the top and bottom by √2.
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…so we could have;
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sin T + cos T = 1

The following is another important formula or identity that must be explored. The triangle below has hypotenuse set to 1;
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…we know that;
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…we can use Pythagoras to form;
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or
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The above is a very important identity to remember, for example lets have θ=30° then we have
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and
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So;
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You can try other examples of your own making sure that you use a calculator if you’re still not convinced.