Reciprocal Trigonometric Functions

We"ve spanned sine, cosine, and also tangent. We"re experts on one tiny piece that trigonometric genuine estate. (Marvin Gardens? Park Place? Boardwalk? Pull the end your syndicate money.) Kudos come you, but there"s more.

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Sine, cosine, and also tangent each have actually a reciprocal function. Reciprocal, reciprocity—think the flipping points over, prefer hamburgers on a grill, pancakes on a griddle, eggs end easy. (When execute we eat?) We currently know that continuous numbers have actually reciprocals (2 and also 1/2 room reciprocals, because that example), yet we can additionally flip ours trig features on their heads.

Cosecant is the reciprocal of sine. That is abbreviation is csc. To identify csc, just flip sin over.

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Secant is the mutual of cosine. Its abbreviation is sec. To recognize sec, simply flip cos over.

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Cotangent is the mutual of tangent. That is abbreviation is cot. To identify cot, simply flip tan over. (Try psychic it all by thinking, "After an night of sin, Joe stretches out on a cot in the backyard and later flips end to obtain a much better tan." Weird, however it works.)

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And no, they don"t sizzle once they"re flipped.

Sample Problem

If a = 8 and b = 15, find the six trig ratios of angle A.

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We"re gonna require that lacking hypotenuse, so an initial use a2 + b2 = c2 to discover c.

82 + 152 = c2

64 + 225 = c2

289 = c2

c = 17

Now let"s plug those sides right into our relationship for sin, cos, and also tan. Psychic SOHCAHTOA. We"re spring at angle A, therefore the opposite next is 8, the surrounding side is 15, and also the hypotenuse is 17. Ready, go.

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Now we take the reciprocals, or upper and lower reversal each duty over.

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Sample Problem

What are the six trig ratios of angle B?

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Once again, we"ll have to track under that hypotenuse before we have the right to take a pilgrimage to TrigVille. Give your buddy Pythagoras a call.

a2 + b2 = c2

We understand the 2 legs the the triangle, for this reason plug "em in because that a and b.

32 + 42 = c2

9 + 16 = c2

25 = c2

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c = 5

Next, discover the sine, cosine, and tangent of edge B.

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To discover the reciprocals, simply flip the fountain over.

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Sample Problem

An isosceles ideal triangle has two legs v a length of 1. If angle A is one of the non-right angles, what are the sine, cosine, tangent, cosecant, secant, and cotangent of angle A?

"Isosceles" looks pretty weird, yet it really just way both legs have actually the specific same length. We don"t have a photo to aid us out, but who demands pictures? We"re math detectives up in here.

All right, therefore we know both legs, which method a = 1 and also b = 1 in ours Pythagorean Theorem. Let"s discover c.

12 + 12 = c2

1 + 1 = c2

c2 = 2

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There"s our hypotenuse. Now, we understand angle A is not the triangle"s ideal angle. It"s one of the other guys, which means its opposite next and nearby side room both 1. Use the trig ratios, and don"t forget come rationalize your denominators.

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When flipping over your fractions to uncover the reciprocals, usage the original fraction (with the pre-rationalized denominator). It will conserve you time and heartache.