Trigonometric: secant θ, cosecant θ, and cotangent θ

In this section we’re going to be looking at the functions secant θ, cosecant θ, cosecant θ. These functions are commonly written as; sec θ, cosec θ or cot θ respectively. Below are the definition for the functions;
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…sec θ is undefined for all values of θ at which cos θ = 0. Also be careful when working with cosine as a reciprocal. Remember cosⁿ ≡ (cos θ)ⁿ for n ∈ z⁺ i.e all positive values but not for n ∈ z^–. For instance cos^-1 does not mean 1/cos θ;
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The second function is cosec θ which is defined as;
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…this is also undefined for values of θ at which sin θ = 0.
…and lastly cot θ;
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…we also know that
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…that must mean that;
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…see the reciprocal article here. So we have;
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Below are some examples using the trigonometric functions explained above.

Examples

You could use your calculator to find cosec θ, sec θ, cot θ. Some calculators have these functions programmed within. Suppose we had to find the following;

• sec 290°
• cot 95°
• sec 45°

Let us start with sec 290°. If sec is not available on your calculator you would first find the cos 240° then divide 1 by the value you find on the calculator;
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Next we shall find cot 95°. We do the same if there is no cot key on your calculator. You first find tan 95° and then divide 1 by it.
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In the following examples we shall find the exact values of the functions (finding values in surd form). Suppose we had to work out;

• sec 210°
• cosec (3π/4)

Let us start with sec 210°. We know that;
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…below we can see that 210° is in the 3rd quadrant;
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…which must mean that;
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…therefore we get;
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…we’re aware that;
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…or we could use the equilateral triangle of side 2 and use Pythagoras to find the value of cos 30°… that is;
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…we can see that;
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…so that must mean;
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Next we can work out;
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…we know that…
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…below we can see that; 3π/4 (135°) is in the 2nd quadrant.
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therefore;
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so we get;
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To find sin (π/4) we can use the right-angled isosceles triangle and then use Pythagoras theorem;
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…therefore the answer is;
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