## Reciprocal Trigonometric Functions

We"ve spanned sine, cosine, and tangent. We"re experts on one little piece of trigonometric genuine estate. (Marvin Gardens? Park Place? Boardwalk? Pull the end your syndicate money.) Kudos to you, but there"s more.You are watching: What is the reciprocal of sin

Sine, cosine, and tangent each have a **reciprocal** function. Reciprocal, reciprocity—think the flipping things over, prefer hamburgers on a grill, pancakes on a griddle, eggs over easy. (When carry out we eat?) We currently know that constant numbers have reciprocals (2 and also 1/2 are reciprocals, because that example), however we can also flip our trig attributes on their heads.

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

**Secant** is the reciprocal of cosine. The abbreviation is sec. To recognize sec, simply flip cos over.

**Cotangent** is the reciprocal of tangent. That is abbreviation is cot. To determine cot, just flip tan over. (Try psychic it every by thinking, "After an evening of sin, Joe stretches the end on a cot in the backyard and later flips end to gain a much better tan." Weird, however it works.)

And no, lock don"t sizzle when they"re flipped.

### Sample Problem

If *a* = 8 and also *b* = 15, uncover the six trig ratios of edge *A*.

We"re gonna need that missing hypotenuse, so an initial use *a*2 + *b*2 = *c*2 to uncover *c*.

82 + 152 = *c*2

64 + 225 = *c*2

289 = *c*2

*c* = 17

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

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

### Sample Problem

What are the 6 trig ratios of edge *B*?

Once again, we"ll should track under that hypotenuse before we deserve to take a expedition to TrigVille. Offer your buddy Pythagoras a call.

*a*2 + *b*2 = *c*2

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

32 + 42 = *c*2

9 + 16 = *c*2

25 = *c*2

*c* = 5

Next, discover the sine, cosine, and tangent of angle *B*.

To uncover the reciprocals, just flip the fractions over.

### Sample Problem

An isosceles appropriate triangle has actually two legs with a length of 1. If edge *A* is among the non-right angles, what space the sine, cosine, tangent, cosecant, secant, and also cotangent of edge *A*?

"Isosceles" looks pretty weird, however it really just way both legs have actually the precise same length. Us don"t have a picture to aid us out, but who demands pictures? We"re *math detectives* increase in here.

All right, therefore we recognize both legs, which means *a* = 1 and *b* = 1 in our Pythagorean Theorem. Let"s discover *c*.

12 + 12 = *c*2

1 + 1 = *c*2

*c*2 = 2

There"s our hypotenuse. Now, we understand angle *A* is *not* the triangle"s appropriate angle. It"s among the other guys, which means its opposite side and nearby side are both 1. Apply the trig ratios, and don"t forget to rationalize her denominators.

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When flipping over her fractions to discover the reciprocals, usage the original portion (with the pre-rationalized denominator). That will conserve you time and heartache.